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PMU-based State Estimation for Hybrid AC and DC Grids

WEI LI

Doctoral Thesis Stockholm, Sweden 2018

KTH Royal Institute of Technology School of Electrical Engineering and Computer Science Department of Electric Power and Energy Systems TRITA-EECS-AVL-2018:23 SE-100 44 Stockholm ISBN 978-91-7729-723-9 SWEDEN Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i elkraftteknik torsdag den 12 april 2018, klockan 10:00, i sal F3, Kungliga Tekniska högskolan, Lindstedtsvägen 26, Stockholm. © Wei Li, April 2018. All rights reserved. Tryck: Universitetsservice US AB

Abstract Power system state estimation plays key role in the energy management systems (EMS) of providing the best estimates of the electrical variables in the grid that are further used in functions such as contingency analysis, automatic generation control, dispatch, and others. The invention of phasor measurement units (PMUs) takes the power system operation and control into a new era, where PMUs’ high reporting rate and synchronization characteristics allow the development of new wide-area monitoring, protection, and control (WAMPAC) application to enhance the grid’s resiliency. In addition, the large number of PMU installation allows the PMU-only state estimation, which is ready to leap forward today’s approach which is based on conventional measurements. At the same time, high voltage direct current (HVDC) techniques enable to transmit electric power over long distance and between different power systems, which have become a popular choice for connecting variable renewable energy sources in distant locations. HVDCs together with another type of power electronic-based devices, flexible AC transmission system (FACTS), have proven to successfully enhance controllability and increase power transfer capability on a long-term costeffective basis. With the extensive integration of FACTS and HVDC transmission techniques, the present AC networks will merge, resulting in large-scale hybrid AC and DC networks. Consequently, power system state estimators need to consider DC grids/components into their network models and upgrade their estimation algorithms. This thesis aims to develop a paradigm of using PMU data to solve state estimations for hybrid AC/DC grids. It contains two aspects: (i) formulating the state estimation problem and selecting a suitable state estimation algorithm; (ii) developing corresponding models, particularly for HVDCs and FACTS. This work starts by developing a linear power system model and applying the linear weighted least squares (WLS) algorithm for estimation solution. Linear network models for the AC transmission network and classic HVDC links are developed. This linear scheme simplifies the nonlinearities of the typical power flow network model used in the conventional state estimations and has an explicit closed-form solution. However, as the states are voltage and current phasors in rectangular coordinates, phasor angle is not an explicit state in the modeling and estimation process. This also limits the linear estimators’ ability to deal with the corrupt angle measurements resulting from timing errors or GPS spoofing. Additionally, it is cumbersome to select state variables for an inherently nonlinear network model, e.g., classic HVDC link, when trying to fulfill its linear formulation requirement. In contrast, it is more natural to use PMU measurements in polar coordinates because they can provide an explicit state measurement set to be directly used in the modeling and estimation process without form changes, and more importantly, it allows detection and correction for angle bias which emerges due to imperfect synchronization or incorrect time-tagging by PMUs. To this end, the state estimation problem needs to be formulated as a nonlinear one and the nonlinear WLS is applied

iv for solution. We propose a novel measurement model for PMU-based state estimation which separates the errors due to modeling uncertainty and measurement noise so that different weights can be assigned to them separately. In addition, nonlinear network models for AC transmission network, classic HVDC link, voltage source converter (VSC)-HVDC, and FACTS are developed and validated via simulation. The aforementioned linear/nonlinear modeling and estimation schemes belong to static state estimator category. They perform adequately when the system is under steady-state or quasi-steady state, but less satisfactorily when the system is under large dynamic changes and the power electronic devices react to these changes. Testing results indicate that additional modeling details need to be included to obtain higher accuracy during system dynamics involving fast responses from power electronics. Therefore, we propose a pseudo-dynamic modeling approach that can improve estimation accuracy during transients without significantly increasing the estimation’s computational burden. To illustrate this approach, the pseudo-dynamic network models for the static synchronous compensator (STATCOM), as an example of a FACTS device, and the VSC-HVDC link are developed and tested. Throughout this thesis, WLS is the main state estimation algorithm. It requires a proper weight quantification which has not been subject to a sufficient attention in literature. In the last part of thesis, we propose two approaches to quantify PMU measurement weights: off-line simulation and hardware-in-the-loop (HIL) simulation. The findings we conclude from these two approaches will provide better guidance for selecting proper weights for power system state estimation.

Sammanfattning Uppskattning av kraftsystemets tillstånd spelar en nyckelroll i energihanteringssystemen (EMS) för att göra den bästa uppskattningen av elektriska variabler i nätet som används i analysfunktioner som beredskapsanalys, automatisk generationskontroll med flera. Uppfinnandet av fasmätningsenheter (PMU) tar drift och kontroll av kraftsystem till en ny era, där PMU:ers höga rapporteringstakt och synkroniseringsegenskaper möjliggör utvecklandet av ny övervakning, skydd och kontroll över stora områden (WAMPAC) för att förhöja nätets elasticitet. Dessutom möjliggör ett stort antal PMU installationer tillståndsuppskattningar enbart baserat på dessa PMU:er, vilket kan förbättra dagens metod som använder sig av konventionella mätningar. Samtidigt gör högspänd likström (HVDC) det möjligt att överföra elkraft över långa avstånd och mellan olika kraftsystem, vilket har blivit ett populärt alternativ för att ansluta förnyelsebara energikällor på avlägsna platser. HVDC-ledningar tillsammans med en annan typ av kraftelektronik-baserade enheter, flexibla AC överföringssystem (FACTS), har visats att framgångsrikt förhöja styrbarheten samt öka kraftöverföringskapaciteten på ett långsiktigt kostnadseffektiv sätt. Med en omfattande integration av FACTS och HVDC överföringstekniker, kommer de nuvarande AC näten att sammanfogas, vilket resulterar i storskaliga hybrida AC och DC nät. Följaktligen behöver kraftsystems estimatorer beakta DC nät/komponenter i sina nätverksmodeller samt uppgradera sina uppskattningsalgoritmer. Den här avhandlingen syftar till att utveckla ett paradigm av att använda data från PMU för att göra tillståndsuppskattningar för AC/DC nät. Den innehåller två aspekter: (i) formulering av tillståndsuppskattningsproblemet och val av passande uppskattnings-algoritm; (ii) utveckla motsvarande modeller, speciellt för HVDC och FACTS. Det här verket börjar med att utveckla en linjär kraftsystemsmodell och tillämpar den linjära viktade minsta-kvadrat-metoden (WLS) för uppskattningslösning. Linjära nätverksmodeller för AC nätet och klassiska HVDC länkar är utvecklade. Den här linjära proceduren förenklar olinjäriteten i den typiska kraftnätverksmodellen som används i konventionella tillståndsuppskattningar och har en explicit lösning på sluten form. Dock, eftersom tillstånden är spännings- och strömfasvektorer i rektangulära koordinater, är fasvinkeln inte ett explicit tillstånd i modelleringsoch uppskattningsprocessen. Det begränsar även de linjära estimatorernas förmåga att hantera de korrupta vinkelmätningarna som är en produkt av tidsfel eller klockförskjutning i GPS:en. Dessutom är det besvärligt att välja tillståndsvariabler för en immanent olinjär nätverksmodell, t.ex. klassisk HVDC-länk, och försöka uppfylla linjära formuleringskrav. Det är däremot mer naturligt att använda PMU mätningar i polära koordinater eftersom de kan tillhandahålla en explicit tillståndsmätning som kan användas direkt i modellerings- och uppskattningsprocessen utan att ändra form, och ännu viktigare, så tillåter den detektering och korrektion för systematiska avvikelser i vinkeln som uppstår på grund av ofullkomlig synkronisering eller felaktig tid-

vi taggning av PMU:er. För detta ändamål behöver tillståndsuppskattningsproblemet formuleras som ett olinjärt problem och olinjär WLS används för lösning. Vi föreslår en ny mätningsmodell för PMU-baserad tillståndsuppskattning som separerar felen som uppstår på grund av modellosäkerhet och på grund av mätbrus så att de kan tilldelas olika vikter separat. Dessutom är olinjära nätverksmodeller för AC nät, klassisk HVDC-länk, så kallad Voltage source converter (VSC)-HVDC, och FACTS utvecklade och validerade via simuleringar. De tidigare nämnda linjära och olinjära modellerings- och uppskattningsschemana tillhör kategorin för statiska tillståndsestimatorer. De fungerar adekvat när systemet är i stabilt, eller kvasi-stabilt, tillstånd, men mindre acceptabelt när systemet är under stora dynamiska förändringar då kraftelektronikenheter regerar på dessa förändringar. Testresultat indikerar att ytterligare modelleringsdetaljer måste inkluderas för att erhålla högre precision under systemtransient dynamik som involverar snabba responser från kraftelektronik. Därför föreslår vi en pseudodynamisk modelleringsmetod som kan förbättra uppskattningsprecisionen under transienter utan att signifikant öka uppskattningens beräkningsbörda. För att illustrera detta tillvägagångssätt är de pseudo-dynamiska nätverksmodellerna för den statiska synkrona kompensatorn (STATCOM), som ett exempel på en FACTS enhet, och VSC-HVDC-länken utvecklade och testade. Genom hela denna avhandling är WLS den huvudsakliga tillståndsuppskattningsalgoritmen. Den kräver en lämplig viktkvantifiering, vilket inte har fått tillräcklig uppmärksamhet i litteraturen. I den sista delen av avhandlingen föreslår vi två metoder för att kvantifiera vikter för PMU-mätningar: off-line simulering och Hardware-in-the-loop (HIL) simulering. De resultat vi erhåller från dessa två tillvägagångssätt kommer att ge bättre vägledning för hur man ska välja lämpliga vikter för uppskattning av kraftsystemstillstånd.

To Meng, Emma, and my parents.

Acknowledgements I would like to firstly express my gratitude to my main advisor Prof. Luigi Vanfretti for your persisting support, guidance and encouragement in my work. You introduced me to the world of research with the master thesis project in 2011. I am extremely grateful that you give me the perfect blend of guidance and independence during this journey. I would like to thank Prof. Joe H. Chow for hosting me at Rensselaer Polytechnic Institute during Jan.-May, 2015. I had a great and fruitful time there and got to know lots of awesome people. Your sincerity and passion for research will continue influencing me later on. The five years of PhD study at KTH have been the most memorable time in my life so far. Electric power and energy systems department has been such a joyful place to live and work. I thank all the colleagues for fun time during lunches and fikas. I thank all my officemates, Yuwa, Francisco, Maxime, Almas, Farhan, Jan, Elis and Evelin for creating a pleasant working atmosphere. I thank Yalin, Meng S. and Zhao for native speaking chats. Special thanks go to Almas and Viktor for helping me set up the hardware test in SmarTS Lab. Thanks also go to the administrators for always being helpful. Finally, I would like to thank my parents for always believing in and supporting me, and my husband Meng for your genuine and everlasting love, as well as my daughter Emma for your limitless trust and beautiful smiles. Wei Li Stockholm, April 2018

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1 Introduction 1.1 Motivating examples . . . . . . . . . . 1.1.1 Power system state estimation 1.1.2 Phasor measurement units . . . 1.1.3 High voltage direct current . . 1.2 Contributions and outline . . . . . . . 1.3 Acronyms . . . . . . . . . . . . . . . .

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2 Power system state estimation overview 2.1 Background . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Power system structure . . . . . . . . . . . 2.1.2 System operating conditions . . . . . . . . . 2.1.3 Power system state estimation . . . . . . . 2.2 PMUs and phasor state estimators . . . . . . . . . 2.2.1 SCADA-based and PMU-based SEs . . . . 2.3 State estimations for HVDC and FACTS . . . . . . 2.4 Static, forecasting-aided, and dynamic SEs . . . . . 2.4.1 Forecasting-aided state estimation . . . . . 2.4.2 Dynamic state estimation . . . . . . . . . . 2.5 Transmission system, and distribution system SEs 2.6 Centralized, distributed, and multi-area SEs . . . .

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3 Conventional state estimations and test systems 3.1 Conventional state estimations . . . . . . . . . . . 3.1.1 Measurement’s distribution: Gaussian . . . 3.1.2 Objective function: weighted least squares . 3.1.3 Numerical solutions . . . . . . . . . . . . . 3.2 Test systems . . . . . . . . . . . . . . . . . . . . .

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Contents 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5

9-bus system . . . . . . . . . . . . . KTH-Nordic 32 system . . . . . . . 6-bus hybrid AC/DC system . . . . VSC-based HVDC transmission link Synthetic measurement generation .

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4 Linear network models 4.1 AC transmission network . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Measurement model . . . . . . . . . . . . . . . . . . . . . 4.2 Classic HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Measurement model for hybrid AC/DC systems . . . . . . 4.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 LSE for the 9-bus system . . . . . . . . . . . . . . . . . . 4.3.2 LSE for the 9-bus hybrid AC/DC system with a classic HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 LSE for the KTH-Nordic 32 system . . . . . . . . . . . . 4.3.4 LSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC link . . . . . . . . . . . . . . . . . . . . . 4.3.5 Effect of less DC measurements . . . . . . . . . . . . . . . 4.3.6 Effect of measurement noise . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Nonlinear network models 5.1 AC transmission network . . . . . . . . . . . . . . . . . 5.2 Classic HVDC link . . . . . . . . . . . . . . . . . . . . . 5.2.1 Network model . . . . . . . . . . . . . . . . . . . 5.2.2 Measurement model for hybrid AC/DC systems . 5.2.3 Considerations for practical application . . . . . 5.2.4 Case study . . . . . . . . . . . . . . . . . . . . . 5.3 VSC-HVDC . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 VSC substation model . . . . . . . . . . . . . . . 5.3.2 VSC control modes . . . . . . . . . . . . . . . . . 5.3.3 Point-to-point VSC-HVDC link model . . . . . . 5.3.4 Case study . . . . . . . . . . . . . . . . . . . . . 5.4 FACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Shunt devices . . . . . . . . . . . . . . . . . . . . 5.4.2 Series devices . . . . . . . . . . . . . . . . . . . . 5.4.3 Case study . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Pseudo-dynamic network modeling and examples 6.1 Pseudo-dynamic concept . . . . . . . . . . . . . . . . . . . . . . .

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Contents 6.2 6.3

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xiii STATCOM model . . . . . . . . . . . . . . . . . . . . . . . VSC-HVDC model . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 VSC substation model . . . . . . . . . . . . . . . . . 6.3.2 Point-to-point VSC-HVDC link model . . . . . . . . Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 STATCOM models’ comparison and validation using PMU data . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 STATCOM model in two test systems . . . . . . . . 6.4.3 VSC-HVDC model . . . . . . . . . . . . . . . . . . . Computation performance . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Quantifying PMU measurement weights 7.1 Quantification approaches . . . . . . . . . 7.1.1 Off-line simulation . . . . . . . . . 7.1.2 Hardware-in-the-loop simulation . 7.2 Results . . . . . . . . . . . . . . . . . . . . 7.2.1 Off-line simulation . . . . . . . . . 7.2.2 Hardware-in-the-loop simulation . 7.3 Summary . . . . . . . . . . . . . . . . . .

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8 Conclusion

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Bibliography

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Chapter 1

Introduction Electricity is so ubiquitous in our daily life that we take it for granted like air we breathe. Electric power system is the network behind to supply, transfer, store and consume electric power. Failure or outage of electric power system can result in tremendous risk to the environmental and public safety. An accurate estimation of such system’s operating states is of critical importance for further tasks like frequency control and load balancing. The development and installation of phasor measurement units (PMUs) help to provide high-resolution, time-synchronized measurement across a large geographic area. Moreover, the unprecedented development in high voltage direct current (HVDC) transmission technology exhibits a great need for highperformance real-time state estimation techniques.

1.1

Motivating examples

This chapter presents the motivation of this thesis work via giving three examples to demonstrate the importance of power system state estimation, the advantage of using PMUs for state estimation, and the necessity of including power electronic-based devices (e.g., HVDCs, FACTS) into the state estimation, respectively. This reflects the three main elements in the thesis title “PMU-based State Estimation for Hybrid AC and DC Grids".

1.1.1

Power system state estimation

Power grids become more complex with increasing electricity consumption and renewables exploitation. For instance, from 2000 to 2016, more than 11,655 megawatts of new generating capacity have been added to the New York State’s electric system, and more than 2,765 MW of transmission capability have been added as well, which is shown in Fig. 1.1. In 2016, the total generating capacity reaches 38,576 MW and the total circuit miles of transmission come to 11,124 miles (≈ 17,902 km). In addition, New York’s largest demand response program, Special Case Resources, is projected to be capable of offering up to 1,248 MW [1]. 1

2

Introduction

Figure 1.1: New Transmission in New York State: 2000-2016 (source [1])

Despite power grids generally are designed to meet demand extremes and handle the worst-case situation, to ensure power system reliability is always a challenge. Transmission system operators (TSOs) or independent system operators (ISOs) have overall responsibility to maintain the short-term balance between production and consumption of electricity. Therefore, they control and monitor grids around the clock in their control rooms. A picture of the control center at the New York ISO is shown in Fig. 1.2. There the computer systems receive 50,000 data points about every six seconds, and operators monitor regional activity on a 2,300-square-foot video wall. Mandatory reliability standards have been put in place for the thousands of entities involved in the operation of the country’s electric systems [2]. Across the world, tasks in power system control rooms are similar: monitoring electricity zooming through the national or regional grid and electric power exchange with neighboring grids. This task is conducted with the help of power system state estimation (PSSE), which provides the optimal estimation of the system’s status based on the collected measurements and the presumed network model. Sequentially, exploiting the estimation results, operators enable to constantly coordinate the supply and demand of electricity to ensure enough electricity is available to keep the lights on without overloading transmission lines. If the system is out of balance or the flow of electricity is interrupted, it can damage equipment or cause power outages.

1.1. Motivating examples

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Figure 1.2: Control center at the New York Independent System Operator: (i) features a 2,300-square-foot video wall—the largest utility installation in North America; (ii) displays more than 3,000 live status points—presenting line flows, limits, transformer loading, voltages, and generator output; (iii) customizes the regional electric system information, weather and lightning-strike data, load forecast, etc. to address system dynamics [3]. In 2003, the worst blackout of the United States started with a sagging power line in Ohio that shorted out after touching a tree branch. A series of human errors and a computer problem plunged about 50 million people into darkness from New York City to Toronto and cost the United States economy about $6 billion [2]. Operators carry out extensive trainings in simulation labs to devise strategies to ensure system reliability when subjected to such faults. Fast and accurate state estimation techniques would enable operators to detect such faults faster and thus react in a more timely fashion.

1.1.2

Phasor measurement units

The performance of power system state estimation relies on the properties of the collected measurements, such as measurement quantities, sampling rate, accuracy and variance, synchronization, etc. At present, phasor measurement units (PMUs) are the most accurate and advanced time-synchronized technology available. They provide voltage and current phasor and frequency information, synchronized with high precision to a common time reference provided by the global positioning system (GPS). Implementing PMUs can enhance the accuracy and computational efficiency of power system state estimation. Figure 1.3 gives an intuitive comparison between the data from a PMU and the data from the traditional supervisory control and data acquisition (SCADA) system. For the past decade, PMUs installation and deployment have dramatically increased. By 2014, the American Recovery and Reinvestment Act (ARRA) investment

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Introduction

Figure 1.3: Data comparison between PMU and SCADA [4]. resulted in total number of installed PMUs to more than 1,100, offering nearly 100 percent observability of the transmission system. Based on a report of North American SynchroPhasor Initiative (NASPI), up till 2017 summer, over 2,500 networked PMUs have been installed and connected in America synchrophasor networks, and this number continues to grow [5]. In China, by the end of 2013, the number of PMUs installed at substations and power plants by the State Grid Corporation of China (SGCC) was 2,027 [6]. The Powergrid of India is installing about 1,700 PMUs covering all 400 kV and above voltage level substations and major generating stations [7]. With the rapid deployment of PMUs, using only PMU data for power system state estimation is feasible and promising. However, more research need to be carried out in order to utilize PMU data in a more efficient and reliable fashion.

1.1.3

High voltage direct current

Presently, alternating current (AC) power technology is dominant in electricity generation, transmission and distribution. For high voltage transmission systems, compared to HVAC, HVDC techniques enable (i) low loss interconnection over large distances, (ii) connection between asynchronous grids, (iii) connection to remote energy resources and loads, and (iv) flexibility to accommodate variable renewable energy. In addition, over a specific distance, called as break-even distance, HVDC line becomes cheaper than HVAC. The break-even distance for overhead lines is around 600 km and for submarine lines it is around 50 km. A report from Bloomberg new energy finance (BNEF) [9] shows the global HVDC capacity has experienced an exponential growth since 2010, and by 2025 the capacity will double the number in 2015. Based on the “Compendium of all HVDC projects 2009” [10] prepared by the SCB4 (HVDC and Power Electronics) committee of CIGRE, the number of HVDC link projects in Africa, Australia and

1.1. Motivating examples

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STAMNÄTET FÖR EL 2017

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Det svenska stamnätet för el består av 15 000 km kraftledningar, 160 transformator- och kopplingsstationer och 16 utlandsförbindelser.

Narvik Ofoten 400 kV ledning 275 kV ledning 220 kV ledning

Rovaniemi

HVDC (likström) Samkörningsförbindelse för lägre spänning än 220 kV Planerad/under byggnad

Røssåga

Vattenkraftstation

Kemi

SVERIGE

Värmekraftstation

Luleå

Vindkraftpark Transf./kopplingsstation Planerad/under byggnad

Uleåborg

Tunnsjødal

FINLAND

Umeå

Trondheim

Vasa

Nea Sundsvall

NORGE

Tammerfors Olkiluoto

Rjukan

Bergen

Forsmark

Rauma

Viborg Loviisa

Åbo

Helsingfors

Oslo

Tallinn

Hasle

Stavanger

Stockholm

(300 kV)

Örebro Kristiansand

ESTLAND Norrköping

Göteborg

LETTLAND

Riga

Oskarshamn Ringhals

DANMARK

Karlshamn

Klaipeda

Malmö

Flensburg Kiel Lübeck Eemshaven

LITAUEN

Köpenhamn

Vilnius

Slupsk Rostock Güstrow 0

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Figure 1.4: Map of the Swedish national grid (source [8])

200 km

6

Introduction

Oceania, Asia, Europe, North America, and South America was 2, 4, 15, 14, 10, and 2 up till 2009; and the number of back-to-back HVDC projects reached 30 in total. This compendium has not been updated since then, while Wikipedia provides a list of HVDC projects in December 2017, which can act as an informal reference for the updated information. A rough count on the HVDC link projects in Africa, Australia and Oceania, Asia, Europe, North America, and South America is 3, 5, 55, 64, 21, and 3; and the number of back-to-back HVDC projects reaches 53 in total [11]. Inside Europe, to allow long-distance transmission for renewables and crossborder exchange of power, HVDC becomes the key to an integrated European power network. For instance, the HVDC links (indicated by purple lines) of Northern Europe are shown in Fig. 1.4. HVDC can provide fast, precise and flexible control of transmission flow, which greatly improves European grid’s reliability, capacity and efficiency. With the extensive integration of HVDC transmission techniques, the present AC dominant networks will merge to large-scale hybrid AC and DC networks. To facilitate meeting this challenge, power system state estimators need to adapt DC grids into the network models and upgrade the estimation algorithms. More research in this field is warranted.

1.2

Contributions and outline

In this section, the main contributions and outline of this thesis are summarized including the publications that each chapter is based upon.

Chapter 2 — Power system state estimation overview This chapter introduces the background and the history of power system state estimation. How to include PMU measurements and consider power electronic-based devices for state estimation is also discussed. In addition, this chapter provides a novel perspective to overview power system state estimation through demonstrating different categorizations. Depending on the timing and evolution of the estimates, state estimation schemes can be classified into static, forecasting-aided, and dynamic state estimations. The target system of the state estimation function decides whether it is a transmission system or a distribution system state estimation. Whether distributing the estimation computation among different geographical areas classifies state estimations into centralized, distributed, and multi-area state estimations.

Chapter 3 — Conventional state estimations and test systems This chapter focuses on the formulation and derivation of state estimation methods, particularly the formulation and solution of the conventional state estimations. In addition, the test systems that are developed and implemented for case studies are introduced. All of the aforementioned contents provide a background for the following chapters.

1.2. Contributions and outline

7

Chapter 4 — Linear network models Using synchronized phasor measurements makes it possible to formulate the state estimation problem as a linear one. This chapter presents the linear network and measurement models for both the AC transmission network and the classic HVDC link. The above results have been published in: • W. Li and L. Vanfretti. Inclusion of Classic HVDC links in a PMU-Based State Estimator. IEEE PES General Meeting, Washington DC, USA, 2014.

Chapter 5 — Nonlinear network models This chapter introduces a novel measurement model for PMU-based state estimation, which separates the network model equations from measurements so that different weights can be assigned to them separately. Corresponding nonlinear network models for the AC transmission network, classic HVDC link, VSC-HVDC, and FACTS devices are presented. The above results have been published in: • W. Li and L. Vanfretti. A PMU-based state estimator considering classic HVDC links under different control modes. Sustainable Energy, Grids and Networks, vol. 2, pp. 69-82, 2015. • W. Li and L. Vanfretti. A PMU-based state estimator for networks containing VSC-HVDC links. IEEE PES General Meeting, Denver, USA, 2015. • W. Li and L. Vanfretti. A PMU-based state estimator for networks containing FACTS devices. IEEE PowerTech, Eindhoven, Netherlands, 2015. Note that the models and case studies for some FACTS devices presented in this thesis have been modified from the corresponding paper.

Chapter 6 —Pseudo-dynamic network modeling and examples This chapter presents the pseudo-dynamic modeling approach that leverages the existing body of the static network model and includes the difference equations that describe system dynamical properties, so that it can improve the state estimation accuracy during transients without significantly increasing its computational burden. Moreover, pseudo-dynamic models for STATCOM and VSC-HVDC are demonstrated as examples. The comparison and validation for the STATCOM model using real PMU data were conducted at Rensselaer Polytechnic Institute under the supervision of Prof. Joe. H. Chow. The above results have been published in: • W. Li, L. Vanfretti and J. H. Chow. Pseudo-Dynamic Network Modeling for PMU Based State Estimation of Hybrid AC/DC Grids. IEEE Access, vol. 6, pp. 4006-4016, 2018.

8

Introduction

Chapter 7 — Quantifying PMU measurement weights In power system state estimation, weighting is a practice of accounting for the confidence in the model and in a measurement. In order to quantify PMU measurement weights, two methods are proposed: off-line simulation and hardware-in-the-loop (HIL) simulation. The latter was conducted in SmarTS Lab at KTH. The above results have been submitted to: • W. Li and L. Vanfretti. Quantifying PMU Measurements Weights for Phasor State Estimation. IEEE Access, submitted, Feb. 2018.

Chapter 8 — Conclusions and future work The thesis is concluded with a summary.

Other publications The following publications are not covered in this thesis, but contain material that motivates the work presented here: • L. Vanfretti, W. Li, A. Egea-Alvarez and O. Gomis-Bellmunt, Generic VSCBased DC Grid EMT Modeling, Simulation, and Validation on a Scaled Hardware Platform, IEEE PES General Meeting, Denver, USA, 2015. • R. Rogersten, L. Vanfretti and W. Li, Towards Consistent Model Exchange and Simulation of VSC-HVdc Controls for EMT Studies through C-Code Integration, IEEE PES General Meeting, Denver, USA, 2015. • L. Vanfretti, N. A. Khan, W. Li, M. R. Hasan and A. Haider, Generic VSC and Low Level Switching Control Models for Offline Simulation of VSC-HVDC Systems, Electric Power Quality and Supply Reliability Conference, Rakvere, Estonia, 2014. • M. R. Hasan, L. Vanfretti, W. Li and N. A. Khan, Generic High Level VSCHVDC Grid Controls and Test Systems for Offline and Real Time Simulation, Electric Power Quality and Supply Reliability Conference, Rakvere, Estonia, 2014. • N. A. Khan, L. Vanfretti, W. Li, and A. Haider, Hybrid Nearest Level and open loop control of Modular Multilevel Converters with N+1 and 2N+1 levels, EPE’14-ECCE Europe, Lappeenranta, Finland, 2014. • R. Rogersten, L. Vanfretti, W. Li, L. Zhang, and P. Mitra, A Quantitative Method for the Assessment of VSC-HVdc Controller Simulations in EMT Tools, IEEE PES ISGT Europe, Istanbul, Turkey, 2014.

1.3. Acronyms

9

• L. Vanfretti, W. Li, T. Bogodorova and P. Panciatici, Unambiguous Power System Dynamic Modeling and Simulation using Modelica Tools, IEEE PES General Meeting, Vancouver, Canada, 2013.

1.3

Acronyms

AC AVM CSC DC DMU DR DSSE DySE EKF FACTS FASE GPS GW HVDC HVAC IGBT IKF ISO LCC LSE MINLP MLE MW NSE PCC PDF PMU PSSE PWM RMS

Alternative current Average value model Current source converter Direct current DC measurement unit Demand regulation Distribution system state estimation Dynamic state estimation Extended Kalman filter Flexible AC transmission system Forecasting-aided state estimation Global positioning system Gigawatt High voltage direct current High voltage alternative current Insulated-gate bipolar transistor Iterative Kalman filter Independent system operator Line commutated converter Linear state estimation Mixed-integer nonlinear problem Maximum likelihood estimate Megawatt Nonlinear state estimation Point of common coupling Probability density function Phasor measurement unit Power system state estimation Pulse width modulation Root mean square

10 RTU SCADA SE SNR SSSC STATCOM SVC TCSC TSO TSSE UKF VSC WAMPAC WLAV WLS

Introduction Remote terminal unit Supervisory control and data acquisition State estimation Signal-to-noise ratio Static synchronous series compensator Static synchronous compensator Static var compensator Thyristor controlled series compensator Transmission system operator Transmission system state estimation Unscented Kalman filter Voltage source converter Wide-area monitoring, protection, and control Weighted least absolute value Weighted least squares

Chapter 2

Power system state estimation overview In this chapter, we firstly describe the background of power system state estimation, particularly its role in power system operation. With the installation of PMUs and unprecedented development of power electronic-based devices, it is inevitable to take them into account for power system state estimation. To meet the diverse needs for state estimation, measurements over consecutive instants or a system model that considers time evolution can be utilized in addition to the static measurement model, leading to the so-called forecasting-aided state estimation and dynamic state estimation, respectively. Furthermore, state estimation is not exclusive for transmission systems any more, but become applicable to distribution systems as well. The architecture of state estimations, i.e., models, formulations, and computation is no longer confined to a centralized methodology; a distributed or hierarchy scheme may be a better choice in some cases. For each of the aforementioned aspects, some related research in literature is discussed. More background on the formulation and derivation of different methods will be presented in next chapter.

2.1

Background

Power system is a gigantic system covering a large geographical area and influencing a tremendous number of population. A stable and resilient power supply is the top fundamental facility for a society.

2.1.1

Power system structure

A typical power system consists of generation, transmission, and distribution systems. Electricity can be generated in different types of power plants depending on the used energy sources, e.g. coal, gas, hydro, wind, and solar, etc. This generated electricity is transported to distribution systems through the transmission system, which connects the producers with the consumers. To minimize power loss during transmission, transmission systems have high voltages. The receiving end of the transmission system is equipped with substations where electricity is transformed 11

12

Power system state estimation overview

into lower voltages for distribution systems. Then the distribution system feeds the electricity to the commercial and residential consumers. Maintaining stability and functionality of such a complex system is the top priority for most of TSOs or ISOs. Unlike water, electricity can not be stored in a bucket for most cases, thus most electricity is used the instant it is created. The delicate art of balancing the grid is to ensure enough electricity is available to keep the lights on without overloading transmission lines.

2.1.2

System operating conditions

According to [12], a system can be characterized as one of the three conditions— normal, emergency, and restorative. A system is said to be in the normal operating condition if both the load and operating constraints are satisfied. In the emergency condition the operating constraints are not satisfied. In the restorative condition, the operating constraints are satisfied but not the load constraints. Even under the normal operating condition, the system may not be secured in a sense that it can hit the operating constraints and degrade to emergency state when a contingency from the predefined critical contingencies list occurs. In order to obtain a comprehensive knowledge of the system operating condition, it is essential to conduct a power system security analysis, which includes monitoring of the system conditions, identification of the operating state, and determination of the necessary preventive actions in case the system state is found to be insecure. As the set of voltage phasors enables to fully specify the system, it is referred to as the static state of the system. The first step is monitoring. Supervisory control systems were initially applied to monitor and control the status of circuit breakers at the substations. And generator outputs and the system frequency were also monitored for purposes of automatic generation control and economic dispatch. Later the augment of real-time systemwide data acquisition capabilities led to the establishment of the first SCADA system. Nowadays, with the proliferation of PMU installations, time-synchronized measurements with higher reporting rate and accuracy are available for most of substations and even for distribution grids. Are these raw measurements enough for the second step—identifying the system operating state? The information provided by the monitoring step may not always be reliable due to errors in measurements, telemetry failures, communication noise, etc. Furthermore, it may not be economically feasible to communicate all possible measurements even if they are available from the transducers at the substations. In those cases, would it be helpful to apply the pre-known network information such as the admittance matrix?

2.1.3

Power system state estimation

Power system static state estimation was initiated by Fred Schweppe in 1960s with the purpose of converting available information (direct meter readings plus other

2.2. PMUs and phasor state estimators

13

information) into an estimate of the present state of the power system [13, 14, 15]. State estimation essentially is a data processing function, or generally speaking a filter. It seeks for the optimal solution based on the redundant measurements and the assumed system model. Introducing the state estimation function broadened the capabilities of the SCADA system, leading to the establishment of the Energy Management Systems (EMS) whose functions such as contingency analysis, automatic generation control, load forecasting and optimal power flow, etc. heavily rely on the state estimation solution. State estimators typically include the following functions: topology processor, observability analysis, state estimation solution, bad data processing, parameter and structural error processing [16]. Each function is an independent field that receives considerable attention from researchers. Moreover, other related research subjects, such as PMUs deployment, calibrating instrument, estimating network model’s parameters, are also active. This thesis mainly contributes to the function of state estimation solution, specifically in the phasor state estimation formulation and network modeling.

2.2

PMUs and phasor state estimators

The term phasor was firstly presented by Charles Proteus Steinmetz in 1893 when he presented a paper on simplified mathematical description of the waveforms of AC electricity [17]. In late 1970s, the symmetrical component distance relay (SCDR) was introduced, which proposed the recursive algorithm of symmetrical component discrete Fourier transform (SCDFT) for measuring the positive sequence voltages and currents accurately and with the response time of one cycle of the fundamental frequency. To spread this phasor calculator across power system for wide-area monitoring, a synchronized clock pulse was used for sampling at different sites. Precise synchronization of sampling clocks became possible with the advent of GPS satellite system [18]. All these techniques eventually led to the development of PMUs in early-1980s by Dr. Arun G. Phadke and Dr. James S. Thorp at Virginia Tech. PMUs can provide voltage and current phasors and frequency information with typical reporting rates as high as 50/60/100/120 f rames/second, and synchronized by the GPS, which makes the wide-area monitoring, protection, and control (WAMPAC) possible. Utilizing PMUs can significantly improve the performance of WAMPAC, including real-time visualization of power systems, advanced early warning system, post-contingency analysis, state estimation, real-time angular, voltage, and frequency control, inter-area oscillation damping, and so on.

2.2.1

SCADA-based and PMU-based SEs

Traditionally, real-time measurements are collected through the remote terminal units (RTUs) installed in the SCADA system. These measurements are usually sampled and collected with a rate of several seconds and they are not synchronized.

14

Power system state estimation overview

In contrast, PMU measurements are GPS synchronized with a rate as high as 50/60/100/120 f rames/seconds. The measured quantities from these two measurement source are different as well. The typical measurement quantities from the SCADA system include active and reactive power flows on transmission lines, active and reactive power injections at buses, and bus voltage magnitudes. For some cases, there are measurements for line current magnitude as well. While the PMU measurement quantities are mainly bus voltage phasors, line current phasors, and frequency. Differences in measurement quantity lead to the differences in the formulation of network models. Generally, the network model for the conventional state estimations is constructed based on the power flow model, which is of high nonlinearity. In contrast, the network models for PMU-based state estimations connect the measurements for voltage and current phasors (in polar coordinates) with states, i.e., voltage magnitude and angle. This relationship possesses much less nonlinearity. Both of the aforementioned network models are nonlinear. But when using only PMU measurements, it is also possible to formulate a linear network model and a linear state estimator [19, 20] by applying the voltage and current phasors in rectangular coordinates, instead of polar coordinates. This design improves the computational efficiency, however, it loses the availability of directly exploiting independent angle information, resulting in difficulties in angle error detection, as well as obstacles for phase angle monitoring. Phase angle information allows early identification of potential problems both locally and regionally [21]. For instance, monitoring angle separation or rate-ofchange of angle separation between two buses or two parts of a grid can help to determine the stress on the system. Moreover, angle information can be used for more accurate detection of nominal transfer capability (NTC) based on thermal, voltage, or stability limitations. Another critical application of phase angle is during restoration. The phase angle value across an opened tie line or an opened circuit breaker would guide an operator in circuit breaker closing. All the aforementioned phase angle monitoring (PAM) functions require accurate phase angle estimation which can be achieved by phasor state estimators where phase angle is formulated as an independent state [22, 23, 24]. Both linear and nonlinear network models are utilized in this thesis. Their advantages and disadvantages will be demonstrated in the following chapters.

2.3

State estimations for HVDC and FACTS

AC has been the preferred global platform for electricity transmission for the past 100 years. But HVAC transmission has some limitations: (i) constraints on the transmission capacity and distance, (ii) the impossibility of directly connecting two AC power networks of different frequencies. With the rapid growth of exploiting variable renewable energy sources and the flourish in access to electricity, new technologies for transmitting power over long distance and between different power systems are expected to grow far beyond their current levels of deployment. The

2.3. State estimations for HVDC and FACTS

15

HVDC techniques fulfill this need. Electronic converters for HVDC are divided into two main categories. Linecommutated converter (LCC) is made with electronic switches that can only be turned on. It used mercury-arc valves until the 1970s, or thyristors from the 1970s to the present day. Voltage source converter (VSC) is made with switching devices that can be turned both on and off. The world’s first commercial HVDC transmission using VSC converters is the“Gotland HVDC Light” which commissions since 1999. VSCs use fully-controlled power devices, such as insulated-gate bipolar transistor (IGBT). LCC-based HVDC, so called the classic HVDC technology is mature and has been used worldwide. It is still irreplaceable for HVDC applications of high power and voltage ratings, which recently can be up to 10 GW , with voltages up to 1, 100 kV . VSC technology can be useful for interconnecting weak AC systems, for connecting large-scale wind power to the grid, or for HVDC interconnections that are likely to be expanded as multi-terminal direct current (MTDC) systems in future. Compared to the classic HVDC, VSC-HVDC technology offers a key advantage of independent control of active and reactive powers, together with additional benefits in control flexibility and reliability [25]. Up to date, VSC-HVDC is widely and effectively applied to interconnected remote generation plants, isolated remote loads, metropolitan areas, and offshore bays, etc [26]. The market for VSC-HVDC is growing fast in Europe, driven partly by the surge of the investment in offshore wind power. Another type of power electronic-based devices that have been proven to enhance controllability and transfer capability is flexible AC transmission system (FACTS) [27, 28]. The concept of FACTS was introduced in 90s and soon was recognized as an indispensable asset to improve transmission quality and efficiency of existing AC grids. FACTS and HVDC can provide both steady state and dynamic control for power systems [29]. For steady state control, FACTS and HVDC can provide voltage regulation, power flow management and control, congestion management, eliminating bottlenecks, and enhancement of transfer capability, etc. For dynamic control, FACTS and HVDC can provide fast voltage support, rapid power flow control and dynamic congestion management, fast controlled voltage and power compensation, power oscillation damping, voltage stability control, and fault ridethrough, etc. With the extensive integration of FACTS and HVDC transmission techniques, the present AC dominant networks will merge to large-scale hybrid AC and DC networks. On the other hand, to efficiently control hybrid AC and DC networks over large geographical areas synchronized measurements are required. An technology innovation project initiated by Bonneville Power Administration (BPA), i.e.,TIP 318: WAMS-Enhanced HVDC Control for Flexible and Stable Grid Operations [30], aims to increase transfer capability of Pacific AC and DC Intertie (PADI) using synchronized wide area measurements. This project points a direction of utilizing simultaneous state information from both AC and DC parts of the network for

16

Power system state estimation overview

control purposes. To obtain the essential awareness of the states, hybrid AC and DC state estimators need to be developed by including FACTS and HVDCs into the network models and upgrading the estimation algorithms. HVDC modeling for power system state estimation dates back to 1980s when [31] first includes a classic HVDC link model into an AC system state estimation. [32] presents a relatively simplified AC/DC converter model and extends it into a MTDC model for state estimations. Both of them are developed for conventional state estimations, thus the usage of SCADA data makes them complex and with high nonlinearity. [33, 34] introduce a basic VSC model and a generic VSC-based MTDC model for state estimation, respectively. [35] combines SCADA data with PMU data for hybrid state estimations for VSC-HVDC links. Previous work on modeling FACTS for state estimation has been conducted in the past decade. [36, 37, 38, 39] present FACTS models for the conventional static state estimations using SCADA data. [40, 41, 42] propose to use PMU data to conduct state estimations for the networks containing FACTS devices. This thesis makes a great effort to develop network models for classic HVDC, VSC-HVDC, and FACTS that are suitable for PMU-based state estimation, in order to achieve the utmost goal of wide-area hybrid AC and DC state estimation. Moreover, as these technologies play an important role in improving system controllability and flexibility, their real-time performance during transients also needs to be monitored for on-line operations. To this end, this thesis presents the pseudo-dynamic modeling approach that can accurately represent devices with dynamical properties in both steady state and transient conditions.

2.4

Static, forecasting-aided, and dynamic SEs

Conventional static state estimations are based on the assumption that the system is operating under normal conditions, known as quasi-static regime, where the system varies smoothly and slowly. Thus, static state estimations construct measurement models based on a single scan of measurements. On the other hand, forecasting-aided state estimations (FASEs) and dynamic state estimations (DySEs) gain redundant information in addition to the measurement model, via applying measurements over consecutive instants and using a system model that considers time evolution, respectively.

2.4.1

Forecasting-aided state estimation

Forecasting-aided state estimation extracts valuable information from a consecutive succession of static states evolving in time. For this reason, it was called tracking static state estimation in early literature. The redundant information can help to enhance network observability analysis, overcome data missing, detect and correct bad data, and process network configuration and parameter errors. The state forecasting process is formulated as a transition model, where the forecasted state is represented by the sum of the estimated state at previous time-step,

2.4. Static, forecasting-aided, and dynamic SEs

17

the trend behavior of the state trajectory, as well as the modeling uncertainty. The measurement model remains the same with the static state estimations. Its objective function endures the weighted least squares (WLS) format while incorporates the forecasted state as an additional set of virtual measurements, weighted according to the forecasting error covariance matrix. Moreover, the forecasted state can be substituted into the measurement model to compute the forecasted measurement. The difference between the forecasted measurement vector and the received measurement vector is defined as the innovation vector [43], which can be used for innovation analysis. High innovation acts as an indicator for possible anomalies, such as bad data, network configuration error, sudden change, network parameter error, etc. The key technique of FASEs is to find the parameters that fit the transition model to the historical data. The FASEs concept was firstly proposed by [44] in 1971. However the static time evolution model, i.e., transition model, was an over simplified one: the most recent estimated state is used as one-step ahead forecasting. Reference [45] developed a more appropriate transition model, in which the Holt’s (two parameters) linear exponential smoothing method is converted into a state space representation. Similar exponential smoothing techniques are also implemented in [46]. Computational intelligence tools such as patten analysis [47], artificial neural networks (ANNs) [48], and fuzzy control methods [49] also gain increasing popularity. A more comprehensive related work can be found in [50].

2.4.2

Dynamic state estimation

System dynamic model describes a system’s behavior during transient periods, and it can be represented by differential equations for continuous systems. As the measurements are sampled discretely, this continuous dynamic model can be discretized to be the network model. This network model uses the system’s dynamic behavior to predict the next time-step state, in contrast, the network model of FASE uses historical data to predict the next time-step state. The term “dynamic estimator” can be traced back to the early 1970s [51]. However, the time update model was oversimplified with a strong assumption that the system was in a quasi steady state, so as to avoid any serious attempt to model the time behavior of the system states. Compared to static state estimations, dynamic state estimations did not receive much attention until the last decade, which was mainly due to four reasons [52]. First, static state estimations were sufficient for most cases in terms of filtering measurements. Second, the objective of dynamic state estimations were not always clear. For instance, in early literature the states for dynamic state estimations were voltage magnitude and angle, the same with the conventional static state estimations. However, for many cases voltage magnitude and angle are not dynamic states, and voltage phasors at different buses are not even independent from each other. Third, the dynamic modeling was not always available and practical. Lastly, the computational burden was significantly large. From the last decade, the term “dynamic state estimation” refers to the estimation

18

Power system state estimation overview

for dynamic states and parameters (e.g., generator rotor angles and generator speed) in power system using an appropriate time update model. The Kalman filter family is the most widespread technique for formulating and solving dynamic state estimations. It provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error [53]. Its objective is to estimate a process by using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of (noisy) measurements. As such, the equations for the Kalman filter fall into two groups: time update equations and measurement update equations. For power system state estimation applications, the time update equations contain critical low frequency electromechanical dynamics. They are responsible for projecting forward the current state and error covariance estimates to obtain a prediction for the next time step. The measurement update equations are formulated in the form of the measurement model that is used for the conventional static state estimations to correct the prediction. For a nonlinear dynamic system, extended Kalman filter (EKF) [54] is often used for solving dynamic state estimation through linearization. To mitigate fistorder approximation errors of the EKF, iterative EKF (IEKF) [55] linearizes the system nonlinear equations iteratively to compensate for the higher-order terms. A derivative-free alternative to EKF is the unscented Kalman filter (UKF) [56]. Its idea is to produce several sampling points (Sigma points) around the current state estimate based on its covariance. Then, it propagates these points through the nonlinear map to get more accurate estimation of the mean and covariance of the mapping results. The dynamic state estimations enable a dynamic view of power system in the control room. The estimated dynamic states by DySE can help to improve the real-time control scheme and provide an initialization for a look-ahead dynamic simulation [54]. In addition, the prediction capacities of DySE automatically ensure system observability even in some stringent condition.

2.5

Transmission system, and distribution system SEs

Most state estimation algorithms are designed for improving the awareness of transmission systems as they establish the backbones of power systems. However, this situation has been gradually changed as more timely and accurate state estimations for distribution systems are needed to facilitate the demand response (DR) and the two-way power flow. Since the last decade, the integration of intermittent renewable energy sources, the market deregulation and demand response, as well as the penetration of electric vehicles have increased the overall energy effectiveness, however, at the same time, they stress the grid, complicate the system operation, and introduce more possibilities for frauds. This condition needs to be improved via distribution automation. State estimation for distribution systems enables to provide an essential awareness of

2.5. Transmission system, and distribution system SEs

19

distribution system operation so as to support the distribution automation. Compared to transmission systems, distribution systems lack proper infrastructure for state estimations. The number of installed telemetered devices is limited. Load data obtained from historical local profile and existing automated meter reading (AMR) devices have limited accuracy. Moreover, three phase imbalance and low X/R ratios complicate the measurement functions, making the decoupled estimator not suitable. Pioneer works on distribution system state estimation (DSSE) were conducted since 1990s [57]. However, research and application in this field were not been truly brought into fruition, probably due to the lack of proper infrastructure. Three types of data sources for the distribution system state estimation are listed below [58]. • Equipment connectivity status from automated mapping and facility management (AM/FM) systems, geographical information system (GIS) and outage management systems (OMS). An interface function has to be implemented to convert these connection “maps” and attribution data from AM/FM/GIS to the operational database structure via common information model (CIM) [59]. • Real-time voltage, current and power flow measurements from distribution automations, SCADA, intelligence electronic devices (IEDs), and PMUs. A big issue for these real-time measurement is the time skew problem. Unsynchronized measurements have to be incorporated to a same time reference. • Customer interval demands and distributed energy resources (DERs) output data from customer information system (CIS) and meter data management system (MDMS). Based on the customer billing data and typical load profiles, customer load curve with stochastic contents are utilized as pseudo-measurement to improve the observability of the estimation. Advanced techniques, such as Gaussian mixture model (GMM) [60] and artificial neural network (ANN) are proposed for modeling the load probability. Moreover, the smart meter data can also be taken as pseudo-measurements due to its less frequent update. Due to the distinct features of distribution systems, its state estimation schemes to some extent are different from transmission system state estimations (TSSEs) in terms of network model and objective function. For instance, in TSSEs, power system state estimation generally assumes the system operates in steady state under balanced conditions. This assumption allows to use the positive sequence equivalent circuit for modeling the entire power system. However, in practice, power flow imbalances often appear in distribution systems. To this end, three-phase state estimators [57, 61] are more suitable to handle unbalanced power flow issues. In addition, due to the high fluctuations of the two-way power flows in distribution systems, a Bayesian network model [62] is proposed for DSSE. The objective function for DSSEs can be formulated as a basic weighted least squares (WLS) problem. Feeder branch currents in rectangular coordinates are chosen as the state variables as it is computationally efficient for radial networks. It

20

Power system state estimation overview

can also be formulated as a constrained optimization problem [63], or a nonlinear optimization problem, such as a hybrid particle swarm optimization problem [64]. Robust state estimations are also applied for distribution systems. To obtain a statistically robust load estimation, machine learning algorithms can also be used [65]. An M-estimator combining WLS and weighted least absolute value (WLAV) is proposed in [66] to suppress the effect of bad data. DSSEs can also be formulated using forecasting-aided or dynamic state estimation techniques, such as iterated Kalman filter (IKF) [67], extended Kalman filter (EKF), or unscented Kalman filter (UKF) [56]. Distributed or multi-area state estimations are also attractive candidates for DSSE solutions.

2.6

Centralized, distributed, and multi-area SEs

Power system state estimation is traditionally formulated and computed in a centralized fashion at regional control centers. While now the deregulation of electricity market requires to monitor the power system over a very large geographical area. In this context, decentralized state estimation schemes can enhance the computational performance and the reliability of the estimation algorithms. But at the same time they require more efficient and reliable communication techniques and need to solve time skewness issue. Both distributed and multi-area state estimations focus on interconnected systems. The difference between them mainly lies on the structure of the state vector [68]: in the distributed state estimations several nodes or areas estimate a common state/parameter vector through local collaborations; while in the multi-area state estimations the measurements of each area only relate to a small part of the whole state/parameter vector. Multi-area state estimations can be formulated as either a hierarchical process [69] or a fully distributed manner [70].

Chapter 3

Conventional state estimations and test systems The previous chapter gives a general overview on state estimation problems. This chapter will focus on the formulation and derivation of state estimation methods, particularly the formulation and solution of the conventional state estimations. In addition, the test systems that are developed and implemented for case studies are introduced. All of the aforementioned contents provide a background for the following chapters.

3.1

Conventional state estimations

Power system state estimation aims to find the best match between the realtime measurements and the system states, i.e. voltage phasors at buses. Thereby, most state estimators firstly formulate the mathematical model that describes the relationship between the system states and the measurements as: z = h(x) + e,

(3.1)

where z ∈ Rm is the measurement vector, x ∈ Rn is the unknown true state vector; m and n are the numbers of measurements and states (m ≥ n), respectively. h : Rn → Rm is a function relating the measured quantities to the state variables, which is called the network model. e ∈ Rm is the unknown measurement error vector. As (3.1) contains measurement vector, it is usually called the measurement/observation model.

3.1.1

Measurement’s distribution: Gaussian

To solve state estimation problems requires selecting an x that makes the observed z most likely to be observed, in other words, to find x that maximizes the likelihood of the observed measurements z, i.e., maximum likelihood estimate (MLE). As different measurement sources can possess different likelihood functions, which are 21

22

Conventional state estimations and test systems

often defined by probability density function (PDF), we need to take measurement’s probability distributions into account when selecting suitable state estimation algorithms. Most literature presumes the measurement noise has a Gaussian distribution, i.e., e = [e1 , e2 , · · · , em ] and ek ∼ N (0, σi2 ), where N (0, σi2 ) is a Gaussian distribution with mean 0 and covariance σi2 , ∀i = 1, 2, · · · , m. For the ease of notation, let R denote the diagonal covariance matrix of the measurement noises, which are assumed to be independent random processes. Thus the measurement z also has a Gaussian distribution with the mean of h(x) and the same covariance matrix R. Then the probability (also called likelihood) of observing z given state x can be computed as: m Y

N (zi |h(x)i , σi2 ) =

i=1

m Y

1 p

i=1

e 2

−

[zi −h(x)i ]2 2σ 2 i

2πσi

,

(3.2)

where h(x)i is the ith element of h(x), σi2 is the variance of the corresponding ith measurement zi . In order to perform the optimization of the above likelihood function, (3.2) usually is rewritten into logarithm as (m ) m m [z −h(x)i ]2 Y X − i 1 1 X [zi − h(x)i ]2 m 2 2σ i p L = ln ln(2π) − ln(σi ), = − − e 2 i=1 σi2 2 2πσi2 i=1 i=1 (3.3) where ln(·) is the natural logarithm. Thus the maximization of the log-likelihood 2 Pm i] function is transformed into the minimization of i=1 [zi −h(x) , which is exactly σi2 the formulation of weighted least squares (WLS) problem [16]. This also implicates that WLS solution is equivalent to the MLE for minimizing an L2 -norm cost function.

3.1.2

Objective function: weighted least squares

Basic WLS as derived in (3.3) serves as the most common objective function for solving static state estimation problems. It is formulated as: m

minimize J(x) = x

1 X [zi − h(x)i ]2 1 = [z − h(x)]T R−1 [z − h(x)]. 2 i=1 σi2 2

(3.4)

In addition to the field measurements, there are two other kinds of measurements, known as the pseudo-measurement and the virtual measurement. Pseudomeasurements are manufactured data, such as generator output or substation load demand, that are based on the historical data or the dispatcher’s objective guesses. Virtual measurements are the information that does not require metering, e.g., zero injection at a switching station. Three kinds of measurements own different variances, particularly the variance of the zero injection is zero given the correct topology information, thereby the covariance matrix could become ill-conditioned. Specially when numerically solving WLS, the ill-condition problem becomes more stringent.

3.1. Conventional state estimations

23

In order to overcome the ill-conditioning, an alternative formulation was proposed in [71] where the virtual measurements are represented by equality constrains. Another method to improve numerical stability is Hachtel’s augmented matrix method [72], where the residuals are defined as independent variables. WLS is the most common method used for power system state estimation owing to its computational efficiency and stability. However, it is sensitive to outliers, and a single outlier can distort the estimation results. To overcome this drawback, robust estimators are proposed, one of which is the weighted least absolute value (WLAV). Their objective functions are mathematically formulated and programmed as a mixed-integer non-linear problem (MINLP) [73].

3.1.3

Numerical solutions

The numerical solution presented herein is based on the basic WLS formulation in (3.4). Its optimal solution can be found when the partial derivative of the objective function equals to zero: ∂J(x) = g(x) = −H(x)T R−1 [z − h(x)] = 0, ∂x

(3.5)

where

∂h(x) ∂x is the Jacobian matrix of the network model h(x). In order to find the root of the nonlinear function g(x) = 0, iterative numerical methods are used. For instance, Newton Method, is a powerful technique for solving equations numerically. For each iteration, the linear approximation equation is formulated as: H(x) =

∂g(xk ) (xk+1 − xk ) = g(xk+1 ) − g(xk ), ∂xk

(3.6)

where k denotes the number of iteration and  T ∂H(xk ) ∂g(xk ) =− R−1 [z − h(xk )] + [H(xk )]T R−1 [H(xk )]. ∂xk ∂xk Generally, the basic WLS ignores the second-derivative terms, namely the partial k) derivative ∂H(x ∂xk . Thus the updating rule in (3.6) is rewritten as: G(xk )∆xk = [H(xk )]T R−1 [z − h(xk )],

(3.7)

where G(xk ) = H(xk )T R−1 H(xk ) denotes the gain matrix and (3.7) is called the normal equation. Ideally, the normal equation can be numerically solved via calculating the inverse of the gain matrix G(xk ). However, as aforementioned, due to the ill-condition of the weight matrix, the gain matrix can also become singular. Therefore, more robust linear algebra algorithms are implemented.

24

Conventional state estimations and test systems

As the gain matrix G(x) is usually a symmetric, positive definite matrix, it can be factorized using the Cholesky decomposition [74]. Cholesky decomposition is a special case for LU decomposition. QR decomposition, which is also known as the orthogonal transformation method [75, 76], is numerically more robust than LU decomposition. Additionally, the hybrid method [77] combines both QR decomposition (orthogonal transformation) and Cholesky decomposition.

3.2

Test systems

There are four test systems—9-bus system, KTH Nordic 32 system, 6-bus system, and VSC-based HVDC transmission link—used in this thesis to validate the proposed network models and state estimation algorithms. They are modified or re-implemented to fit various simulation needs for different case studies.

3.2.1

9-bus system

WSCC 3-generator 9-bus test system is already available in Power System Analysis Toolbox (PSAT) [78]. Its one-line diagram is shown in Fig. 3.1. This 9-bus test system is modified and applied for three case studies. • Linear network model of classic HVDC in Chapter 4, where a classic HVDC link replaces the original AC line between bus 7 and bus 8. • Nonlinear network models of FACTS in Chapter 5, where different FACTS devices are placed in different positions according to their specific functions. Generally, shunt devices are installed at bus 8, and series devices are installed on the line between bus 5 and bus 4. • Pseudo-dynamic model of STATCOM in Chapter 6, where a STATCOM is installed at bus 8.

Bus2

Bus7

Bus8

Bus5

Bus9

Bus6 Bus4 Bus1

Figure 3.1: 9-bus test system

Bus3

3.2. Test systems

3.2.2

25

KTH-Nordic 32 system

The KTH-Nordic 32 system is a conceptualization of the Swedish power system and its neighbors. Its predecessor is the CIGRE “Nordic 32A” test network developed by K. Walve [79] and a system data set was proposed by T. Van Cutsem [80]. In [81] some adjustments to the system model and its parameters were made, since then the model is referred to as the KTH-Nordic32 system. For more details, the reader is referred to [81]. This system has already been implemented in PSAT by Y. Chompoobutrgool and its one-line diagram is shown in Fig. 3.2. This KTH-Nordic 32 test system is modified and applied for three case studies. • Linear network model of classic HVDC in Chapter 4, where a classic HVDC link replaces one of the original AC lines between bus 38 and bus 40. • Nonlinear network models of classic HVDC in Chapter 5, where a classic HVDC link replaces the original AC line between bus 36 and bus 41. • Pseudo-dynamic model of STATCOM in Chapter 6, where a STATCOM is installed at bus 43.

3.2.3

6-bus hybrid AC/DC system

The 6-bus hybrid AC/DC test system is designed with parallelized AC line and DC link so as to improve its robustness under stringent perturbations. Its one-line diagram is shown in Fig. 3.3. This test system is applied for two case studies. • Nonlinear network model of classic HVDC in Chapter 5, where the classic HVDC link is placed between bus 3 and bus 4. This model is implemented in PSAT where the classic HVDC link model is available. • Nonlinear network model of VSC-HVDC in Chapter 5, where the VSC-HVDC link is placed between bus 3 and bus 4. This model is implemented in matlab/Simulink where the VSC model is available.

3.2.4

VSC-based HVDC transmission link

There are many VSC-HVDC simulation models proposed in literature. However, in order to make the test system accessible by other researchers, the VSC-Based HVDC Transmission Link model provided by matlab R2013b/Simulink is used to generate the synthetic measurements for the case study in Chapter 6. A detailed description of the model and control strategy can be found in [82].

26

Conventional state estimations and test systems

G19

G9

51

EQUIV.

G2

SL 34

21

23

35

22

24

52

G20

G4 25

G10

G3

G1 G11

G5 26

37

G8

36

NORTH

G12 33

32

38

39

G14

G13

41

40

CENTRAL

43

G7

42

45

48 29

30

G6 G17 49

27

31

28

G16

50

G18

G15

44

47

SOUTH

Figure 3.2: KTH-Nordic 32 test system [81]

46

400 kV 220 kV 130 kV 15 kV

3.2. Test systems

27 Busl

Bus3

Bus4

us

Bus2

us6

Figure 3.3: 6-bus test system

Figure 3.4: VSC-Based HVDC Transmission Link model [82]

3.2.5

Synthetic measurement generation

In all the case studies, synthetic measurements used for off-line state estimations are obtained by running the time-domain simulations of the test systems in PSAT or matlab/Simulink, depending on where the test system is implemented. The simulation results are then re-sampled with the rate of 20 ms to imitate the PMU data. Unless otherwise stated, for off-line state estimations, no measurement noise is added because we focus on investigating the influence of network model on estimation accuracy. Moreover, all the weights for network equations and measurements are assumed to be 1, and full measurement observability is satisfied so as to avoid the influences of weighting and measurement redundancy on estimation results.

Chapter 4

Linear network models Using synchronized phasor measurements allows to formulate the state estimation problem as a linear one. This chapter presents the linear network and measurement models for both AC transmission network and classic HVDC link. As introduced in Chapter 3, measurement model for the conventional state estimations is generally formulated as z = h(x) + e.

(4.1)

Conventional state estimations are performed using measurements such as bus voltage magnitude, active power flow and injection, reactive power flow and injection, current flow, etc. And the state vector includes bus voltage magnitude and angle. Therefore, the network model h(x) is a nonlinear function based on the power flow model. In contrast, as PMUs can measure voltage and current phasors directly, it is possible to formulate a linear network model using Kirchhoff’s circuit law. But the state variables, usually bus voltage phasors, need to be adjusted in rectangular coordinates [19]. The linear measurement model for PMU-based state estimations is given by z = Ax + e, m×n

(4.2)

where A ∈ R is a constant matrix constructed based on Kirchhoff’s circuit law. With the proposed linear measurement model, the basic WLS objective function (3.4) becomes 1 (4.3) minimize J(x) = (z − Ax)T R−1 (z − Ax). x 2 Differently from iteratively solving nonlinear WLS problems, linear WLS problems have explicit closed-form solutions. The Moore-Penrose inverse, Cholesky decomposition, QR decomposition, singular value decomposition (SVD) are applicable to solve linear WLS [83, 84], depending on how well-conditioned the matrices A and R are. This chapter will focus on constructing matrix A from physical models of AC transmission network and classic HVDC link [85]. Sequentially their linear measurement models for static PMU-based state estimation are formulated. 29

30

4.1 4.1.1

Linear network models

AC transmission network Network model

An AC transmission network is a collection of AC lines/branches that can be represented by π-models. Similarly to [22, 23], an AC transmission line comprises a line with series admittance and shunt admittance, as well as a transformer. All these components are enough to formulate the AC network model, which describes the relation between complex voltages at buses and complex currents flowing through the lines adjacent to these buses. This model is similar to those in [16, 78], but taking the shunt admittances and transformers into consideration.

Vf

Vx If

Ix

It

Vt

Figure 4.1: An AC transmission line model [16, 78] As shown in Fig. 4.1, the subscript f denotes the bus where current flows from (i.e., sending end) and t is the bus where current flows to (i.e., receiving end). The ith line is represented by a series admittance yi and shunt admittance yi0 in per Vr transformer is represented by the off-nominal tap ratio a V: i1. In the unit. An ideal case of phase shifting transformers, a is a complex number. Consider a fictitious + bus x between the ideal transformer and the line+ series admittance, thus, for the assumed current directions, we have Rectifier

Inverter 1 V˜x = V˜f , I˜x = a∗ I˜f . a

The current I˜x and I˜t are given by 1 1 I˜x = −yi V˜t + (yi + yi0 )V˜x , I˜t = (yi + yi0 )V˜t − yi V˜x . 2 2 Substituting for I˜x and V˜x , we have 1 1 a∗ I˜f = −yi V˜t + (yi + yi0 )V˜f a 2 1 1 I˜t = (yi + yi0 )V˜t − yi V˜f . 2 a

4.2. Classic HVDC link

31

Writing the above equations in a matrix form "

I˜f I˜t

yi + 12 yi0 a2 − yai



#

= |

"

− ay∗i



yi + 12 yi0 {z }

V˜f V˜t

# .

(4.4)

YAC

More generally, ˜I = YAC V. ˜

(4.5)

Matrix YAC is essentially an admittance matrix that connects the bus voltage phasors with the measured line current phasors. Note that YAC is only part of the coefficient matrix A in the measurement model as there are also measurements of voltage phasors.

4.1.2

Measurement model

For the AC-only state estimations, the state vector contains complex voltage phasors at all buses while the measurement vector contains both current and voltage phasors. Thus the measurement model of an AC network is given by "

zV˜ zI˜

#

"

" # # e Y1 ˜ V ˜ + = V , YAC eI˜ | {z }

(4.6)

A

where   Y1 =  



1 ..

 . 

. 1

Note that the voltage measurements are usually not available for all the buses, thus matrix Y1 usually has less rows than columns.

4.2 4.2.1

Classic HVDC link Network model

A simplified classic HVDC link model is shown in Fig. 4.2. It integrates DC voltages and currents with AC voltages at the terminal buses, which can be measured by PMUs. This model avoids using active and reactive power variables to construct relations.

Vf

Vx If

Ix

It

32

Vt

Linear network models

Vi

Vr +

+

-

-

Rectifier

Inverter

Figure 4.2: A simplified classic HVDC link model [78] The subscript r refers to the rectifier terminal of the classic HVDC link and i refers to the inverter terminal. The network equations are given by [78]: √

Vrdc = 0.995 ∗ 3√π 2 ar |V˜r | cos α − π3 Xr Idc , Vidc = 0.995 ∗ 3 π 2 ai |V˜i | cos δ − π3 Xi Idc , Idc = R1dc (Vrdc − Vidc ),

(4.7)

where Vrdc , Vidc , and Idc refer to the rectifier side DC voltages, inverter side DC voltages, and DC currents, respectively; |V˜r | and |V˜i | are the AC voltage magnitudes at the rectifier terminal bus and at the inverter terminal bus, respectively; ar and ai are the tap ratios; α and δ are the firing angle and extinction angle; Xr and Xi are the transformer reactances; Rdc is the resistance of the DC connection. For the hybrid AC/DC state estimation, in addition to the AC bus voltage phasors, the rectifier side DC voltage, inverter side DC voltage, DC current and products |V˜r | cos α and |V˜i | cos δ are also state variables. Choosing |V˜r | cos α and |V˜i | cos δ as state variables is based on the context of developing a linear model. Therefore, the linear model of (4.7) is formulated as follows 

√



 0.995 ∗ 0     0 =  0 | 

3 2 π ar

− π3 Xr

−1 √

0.995 ∗

3 2 π ai

−1 1 Rdc

{z

YDC

− R1dc



     − π3 Xi     −1 }

|V˜r | cos α |V˜i | cos δ Vrdc Vidc Idc

    .   

(4.8) Since V˜r and V˜i are state variables of the AC network, they can be obtained after every estimation computation. Then, cos α and cos δ can be derived.

4.2.2

Measurement model for hybrid AC/DC systems

The measurement model of a hybrid AC/DC system combines the AC network model and the classic HVDC link model as

4.3. Case study



zV˜

  z|V˜r | cos α   z˜  |Vi | cos δ   zV rdc   zV idc   zIdc    zI   0   0  0 | {z

33





  Y1                =             YAC    | }

 1 1 1 1

YDC {z

A

 ˜ V   ˜ r |cos α   |V    |V   ˜ i |cos δ    Vrdc  1    Vidc  Idc  | {z States }

M easurements



eV˜

  e|V˜r | cos α   e˜  |Vi | cos δ     eV rdc     eV idc +   eIdc       eI   edc1  }  edc2  edc3 | {z 

Residuals

                   }

(4.9)

4.3

Case study

The 9-bus system, the KTH-Nordic 32 system and their modified hybrid AC/DC systems are utilized for this case study. The 9-bus hybrid AC/DC system replaces the original AC line between bus 7 and bus 8 in the 9-bus system with a classic HVDC link. The KTH-Nordic 32 hybrid AC/DC system replaces the original AC line between bus 38 and 40 in the KTH-Nordic 32 system with a classic HVDC link. Other parameters of the hybrid AC/DC systems remain the same as in their original AC systems. Unless otherwise stated, it is assumed that PMUs are installed at all the buses and the DC measurements (zV rdc , zV idc , zIdc , cos α and cos δ) are sampled synchronously and timestamped by GPS at the same rate as the PMU data. Estimation results are shown in such sequence: 9-bus system, 9-bus hybrid AC/DC system, KTH-Nordic 32 system, and KTH-Nordic 32 hybrid AC/DC system. For each case, we show the results from two perspectives: all quantities at one time instant (snapshot), and one quantity in a time series. Sequentially, the effect of having less DC measurements and the effect of measurement noises on state estimation performance are investigated.

4.3.1

LSE for the 9-bus system

The state vector for the 9-bus AC system is organized as follows xAC = [V˜1 V˜2 V˜3 V˜4 V˜5 V˜6 V˜7 V˜8 V˜9 ]T . The time domain simulation is performed in PSAT to generate the synthetic measurements for 10 s, and a 0.1 p.u. (10%) load increase is applied at bus 8 at t = 2 s .

34

Linear network models

Figures 4.3 and 4.4 show the estimation results for a single snapshot and for a time series, respectively. In the simulation environments, the true values of the states are known. Consequently, the estimation error (e.g. |V˜ |est − |V˜ |true ) and estimation residual (e.g. |V˜ |est − |V˜ |meas ) are equivalent. For each subfigure, the true value, the measurement and the estimated result are shown on the top and the estimation error/residual is placed on the bottom. The estimation errors/residuals for the 9-bus system are lower than 10−13 p.u. (or deg).

1 0.95 1

2

3

4

−16

Error (p.u.)

6

x 10

5 6 Bus No.

Vmag−true Vmag−meas Vmag−est 7 8 9

θ (deg)

10 0

1 Vmag−residual−error

4 2 0 1

2

3

4

5 6 Bus No.

7

8

2

3

4

x 10−14

5 6 Bus No.

7

8

9

Vang−residual−error 0.5 0 1

9

˜ at all buses (a) |V|

Vang−true Vang−meas Vang−est

−10 1 Error (deg)

|V| (p.u.)

1.05

2

3

4

5 6 Bus No.

7

8

9

(b) θ at all buses (bus 7 is set as the reference)

Figure 4.3: LSE for the 9-bus system in a single snapshot

1.01 0

100 −15

Error (p.u.)

2

x 10

1 0 0

100

200 300 400 500 time step Er roVmag−residual−error r (d e g) 200 300 400 500 time step

˜ at bus 8 (a) |V|

θ (deg)

|V| (p.u.)

1.015

−2.5

Vang−true Vang−meas Vang−est

−3 −3.5 0 5

Error (deg)

Vmag−true Vmag−meas Vmag−est

1.02

100

x 10−13

200 300 time step

400

500

Vang−residual−error 0 −5 0

100

200 300 time step

400

500

(b) θ at bus 8 (bus 7 is set as the reference)

Figure 4.4: LSE for the 9-bus system in multiple snapshots

4.3.2

LSE for the 9-bus hybrid AC/DC system with a classic HVDC link

The state vector for the 9-bus hybrid AC/DC system is organized as follows  T xAC/DC = xAC |V˜7 | cos α |V˜8 | cos δ V7dc V8dc Idc . The same test scenario in Section 4.3.1 is implemented herein. Figures 4.5 and 4.6 show the estimation results.

4.3. Case study

35 1.05 Hvdc−true Hvdc−meas Hvdc−est

0.95

(p.u.)

1 Vmag−true Vmag−meas Vmag−est

0.9 0.85 1

2

3

4

6

7

8

0.95 1 Vrdc −13 x 10 1

9

Bus No.

−16

x 10

6 Error (p.u.)

5

1

Vmag−residual−error 4 2

Error (p.u.)

|V| (p.u.)

1.05

2

3

4

5

6

7

8

2.5

3 Idc

Hvdc−residual−error 1 Vrdc

9

Bus No.

˜ at all buses (a) |V|

2 Vidc

0.5

0

0 1

1.5

1.5

2 Vidc

2.5

3 Idc

(b) DC state variables (before the perturbation)

Figure 4.5: LSE for the 9-bus hybrid AC/DC system with a classic HVDC link in a single snapshot 0.988

Vmag−true Vmag−meas Vmag−est

0.886 0.884

0.984 0.982 0.98 0

0.882 0

100

200

300

400

Hvdc−true Hvdc−meas Hvdc−est

0.986

Vidc (p.u.)

|V| (p.u.)

0.888

100

500

2

Error (p.u.)

Error (p.u.)

Vmag−residual−error

500

−10

10

1

Hvdc−residual−error

−20

10

0

400

10

x 10

3

0

300

0

−16

4

200

time step

time step

100

200

300

400

0

100

200

300

400

500

time step

500

time step

(b) DC voltage on the inverter side (close to bus0.915 8) Hvdc−true Hvdc−meas Hvdc−est

1.0001 1

I

dc

(p.u.)

1.0001

0.913 0.912

1 0.9999

cosα−ture cosα−meas cosα−est

0.914 cosα

˜ at bus 8 (a) |V|

0

100

200

300

400

500

0.911 0

100

400

500

10

Error

Error (p.u.)

300

0

−5

10

−10

10

Hvdc−residual−error

−15

10

200

time step

time step

0

100

200

300

time step

(c) DC current

400

500

−10

10

cosα−residual−error

−20

10

0

100

200

300

400

500

time step

(d) cos α

Figure 4.6: LSE for the 9-bus hybrid AC/DC system with a classic HVDC link in multiple snapshots In Fig. 4.5b, the DC states are computed before applying the perturbation, and thus the estimated states have small residuals in the order of 10−14 p.u. However,

36

Linear network models

as shown in Figs. 4.6b, 4.6c, and 4.6d, when the system is subject to a perturbation, the estimation residuals for the DC states increase owing to the limitation of the static network model. The static state estimator does not include a dynamic DC link model, thus its static network equations for the classic HVDC link may not hold during system dynamics, especially when its control scheme is active. This can also be confirmed by noting that the estimation residuals decrease as the system’s oscillations decay. Regardless, the static estimation residuals for the DC states are still within an acceptable range. For AC states, the hybrid AC/DC state estimator performs as accurate as the AC-only state estimator.

4.3.3

LSE for the KTH-Nordic 32 system

The state vector for the KTH-Nordic 32 system is organized as follows xAC = [V˜1 V˜2 . . . V˜51 V˜52 ]T . The time domain simulation is performed in PSAT to generate the synthetic measurements for 10 s, and a 0.5 p.u. (10%) load increase is applied at bus 40 at t = 2 s. Figures 4.7 and 4.8 show the estimation results, which remain the same accuracy as that in the 9-bus system. Vmag−true Vmag−meas Vmag−est

1.1 1 0.9 0.8 0

10

20

30

40

50

Bus No.

100

θ (deg)

|V| (p.u.)

1.2

0 −50 −100 0

1

10

20

30

Bus No.

˜ at all buses (a) |V|

40

50

Error (deg)

Error (p.u.)

4

Vmag−residual−error

0.5 0

20

30

40

50

−14

x 10

1.5

0

10

Bus No.

−15

2

Vang−true Vang−meas Vang−est

50

x 10

Vang−residual−error

2 0 −2 −4 0

10

20

30

40

50

Bus No.

(b) θ at all buses (bus 38 is set as the reference)

Figure 4.7: LSE for the KTH-Nordic 32 system in a single snapshot

4.3.4

LSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC link

The state vector for the KTH-Nordic 32 hybrid AC/DC system is organized as follows xAC/DC = [xAC |V˜38 | cos α |V˜40 | cos δ V38dc V40dc Idc ]T . The same test scenario in Section 4.3.3 is applied here. Figures 4.9 and 4.10 show the estimation results. Observe that in Fig. 4.10d the true values of cos α are constant

4.3. Case study

37 −19 Vmag−true Vmag−meas Vmag−est

0.9 0.895

θ (deg)

|V| (p.u.)

0.905

Vang−true Vang−meas Vang−est

−19.5

0.89 0.885 0

100

200

300

400

−20 0

500

100

Error (deg)

Error (p.u.)

1 Vmag−residual−error

1.5 1 0.5 0

0

300

400

500

−13

−15

x 10

2

200

time step

time step

100

200

300

400

x 10

Vang−residual−error

0.5 0 −0.5 −1 0

500

100

time step

200

300

400

500

time step

˜ at bus 40 (a) |V|

(b) θ at bus 40 (bus 38 is set as the reference)

Figure 4.8: LSE for the KTH-Nordic 32 system in multiple snapshots because the classic HVDC link’s control scheme is active, however its estimated values have a variation over time. This is because (i) this linear static model lacks representation of the classic HVDC link’s dynamics as explained in Section 4.3.2, especially for its control scheme; (ii) cos α is not an independent state in this linear state estimator and it is derived from the estimated states |V˜r | cos α and V˜r , thus their estimation accuracies will directly influence cos α. The same condition also applies to cos δ. Vmag−true Vmag−meas Vmag−est

1.1 1 0.9 0.8 0

10

20

30

40

(p.u.)

|V| (p.u.)

1.2

1 Vrdc

50

Bus No. −15

Vmag−residual−error

1

Error (p.u.)

Error (p.u.)

4

1.5

1.5

2 Vide

2.5

3 Idc

−12

x 10

2

Hvdc−true Hvdc−meas Hvdc−est

2.65 2.6 2.55 2.5 2.45

x 10

Hvdc−residual−error

3 2 1

0.5 0

0

0

10

20

30

Bus No.

˜ at all buses (a) |V|

40

50

1

Vrdc

1.5

2

2.5

Vide

3 Idc

(b) DC states (before the perturbation)

Figure 4.9: LSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC link in a single snapshot

4.3.5

Effect of less DC measurements

From (4.9) we can observe that as long as two out of five DC measurements (zV rdc , zV idc , zIdc , zcos α and zcos δ ) and the AC terminal voltage magnitudes |V˜r | and |V˜i | are provided, the other three remaining DC states can be computed. This brings a

38

Linear network models

|V| (p.u.)

0.845

Vidc (p.u.)

2.42 Vmag−true Vmag−meas Vmag−est

0.85

2.4

2.36 0

0.84 0

100

200

300

400

300

400

500

time step 3

−15

x 10

2 1

Error (p.u.)

Hvdc−residual−error Vmag−residual−error

Error (p.u.)

200

−3

x 10

2 1 0

0 0

100

500

time step 3

Hvdc−true Hvdc−meas Hvdc−est

2.38

100

200

300

400

0

100

200

300

400

500

time step

500

time step

Idc (p.u.)

Hvdc−true Hvdc−meas Hvdc−est

2.4 2.35 2.3 0

100

200

300

400

500

cosα

(b) DC voltage on the inverter side (close to bus 40) 0.9975

˜ at bus 40 (a) |V| 2.45

cosα−ture cosα−meas cosα−est

0.997 0.9965 0.996 0

100

x 10

400

500

x 10

1

cosα−residual−error

Hvdc−residual−error 2

Error

Error (p.u.)

300

−3

−6

3

200

time step

time step

0.5

1 0 0

100

200

300

time step

(c) DC current

400

500

0

0

100

200

300

400

500

time step

(d) cos α

Figure 4.10: LSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC link in multiple snapshots

great advantage that only few DC measurements are needed during normal operating conditions as AC measurements can be obtained using PMUs. When a DC measurement is lost, the corresponding rows in the measurement vector, in the residual vector, and in the A matrix need to be removed. As long as the number of rows is equal or larger than the number of columns in A, all the states can be solved out through the WLS. The more rows that A has, the higher redundancy that the estimation owns. The same test scenario in Section 4.3.2 is applied herein. Only the DC measurements of cos α and cos δ are provided. As shown in Fig. 4.11, the residuals of the DC voltages and DC current increase while the residuals of cos α decrease. Even with a limited number of DC measurements it is still possible to obtain acceptable estimation results for the whole AC/DC grid.

4.3. Case study

39 0.99

1.1

p.u.

Vidc (p.u.)

Hvdc−true Hvdc−est

1.05 1 0.95 1

1.5

Vrdc

2

2.5

0.98 0.975 0

3

Vide

Hvdc−true Hvdc−est

0.985

100

200

300

400

500

time step

Idc

−9

1.5

x 10

0

10

Hvdc−residual−error Error (p.u.)

Error (p.u.)

Hvdc−residual−error 1 0.5 0

−5

10

−10

1

1.5

2

Vrdc

2.5

10

3

Vide

0

100

Idc

200

300

400

500

time step

Hvdc−true Hvdc−est

1.02 1.01

cosα−ture cosα−est

0.914 cosα

Idc (p.u.)

(a) DC state variables in a single snapshot (b) DC voltage on the inverter side (close to bus 8) (before the perturbation) 1.03 0.915

1

0.913 0.912

0.99 0

100

200

300

400

0.911 0

500

100

300

400

500

−11

0

10

1.66

x 10

Hvdc−residual−error

cosα−residual−error

1.655 Error

Error (p.u.)

200

time step

time step

−5

10

1.65 1.645

−10

10

0

100

200

300

400

500

1.64 0

time step

(c) DC current

100

200

300

400

500

time step

(d) cos α

Figure 4.11: Effect of less DC measurements: LSE for the 9-bus hybrid AC/DC system with a classic HVDC link (no measurements of Vrdc , Vidc , and Idc are provided)

4.3.6

Effect of measurement noise

Gaussian white noise is added to the synthetic measurements used for the state estimation. The signal-to-noise ratio per sample is 75 dB and the same test scenario in Section 4.3.3 is applied here. Figure 4.12 shows the estimation residuals increase to the order of 10−4 compared to 10−15 achieved when without noise. Nevertheless, this estimation accuracy is acceptable for the signal-to-noise ratio applied.

40

Linear network models

Vmag−true Vmag−meas Vmag−est

1.1 1 0.9

100

θ (deg)

|V| (p.u.)

1.2

Vang−true Vang−meas Vang−est

50 0 −50

0.8 0

10

20

30

40

−100 0

50

Bus No.

10

20

30

40

50

Bus No.

−4

0.02

Vmag−residual−error 1

0 0

10

20

30

Bus No.

˜ at all buses (a) |V|

40

50

Error (deg)

Error (p.u.)

x 10

Vang−residual−error 0.01 0 −0.01 −0.02 0

10

20

30

40

50

Bus No.

(b) θ at all buses

Figure 4.12: Effect of measurement noise: LSE for the KTH-Nordic 32 system in a single snapshot

4.4

Summary

A linear PMU-only state estimator for hybrid AC/DC systems has been introduced in this chapter. Using Kirchhoff’s circuit laws, the proposed linear AC network model simplifies the nonlinearities of the typical power flow network model used in the conventional state estimations. A simplified classic HVDC model is reconstructed into a linear formulation. All the AC and DC states are considered simultaneously for solving the linear least squares problem. After presenting the network and measurement models of the hybrid AC/DC state estimator, case studies are performed. It shows when the system is experiencing transient dynamics, the estimation results for the DC states are not as good as that during normal operating conditions. This is mainly attributed to the linear and static characteristics of the proposed classic HVDC link model. Finally, the effect of having less DC measurements and the effect of measurement noise are discussed.

Chapter 5

Nonlinear network models In the last chapter, we have observed the limitation of using linear state estimators, which lies on the restraints of restructuring a nonlinear network model into linear equations. Hence, this chapter will develop nonlinear network equations for PMUbased state estimations. For the conventional state estimations, the network model h(x) represents the physical relationships between the measurements from SCADA system and the desired unknown states under system steady state. Equation (3.1) assumes that h(x) is well-developed and it perfectly reflects the system’s behavior. However, this ideal case is rarely true in real life as the system always operates around the steady-state, which is defined as the normal operating condition, resulting in a small deviation between the real value of the states and the value from the model. This deviation is the modeling uncertainty and it reflects the model’s limitations to capture the reality. Due to the lack of representation of the modeling uncertainty in (3.1), a novel measurement model for PMU-based state estimations is proposed [86, 87, 88]. It accounts for both modeling uncertainty (due to modeling imperfections) and measurement noise (due to instrumentation and environmental noise, communication noise, etc). This measurement model is formulated as: " # " # u f (x) e= = , (5.1) v x−z where u ∈ Rn and v ∈ Rm are the modeling uncertainty and the measurement noise, respectively. They are independently distributed from each other and E(uvT ) = 0. Lastly, f (x) ∈ Rn is the restructured network model. Accordingly, the objective function of the basic WLS (3.4) is adjusted to minimize x

n m 1  X 1 2 X 1 2 ui + v , 2 i=1 Qii Rjj j j=1

(5.2)

where both u and v are assumed to have a Gaussian distribution, i.e., u ∼ N (0, Q) 41

42

Nonlinear network models

and v ∼ N (0, R). Qii is the ith diagonal element of the modeling uncertainty’s covariance matrix and Q = E(uuT ); Rjj is the jth diagonal element of the measurement’s covariance matrix and R = E(vvT ). In order to solve the above WLS problem, the normal equation (3.7) used for iteratively updating the state variables is adjusted to G(xk )∆xk = [H(xk )]T We(xk ),

(5.3)

where W is the weighing matrix with Qii s and Rjj s on its diagonal, and the gain matrix G(xk ) = [H(xk )]T WH(xk ). As described by (5.1), the network model equations are separated from the measurements. The upper half of the error vector, i.e., u, represents the modeling errors of the network equations, thus weights based on the confidence in the model’s accuracy are individually assigned to them. For the lower half of the error vector, i.e., v, the measurement errors are for the PMU measured quantities: bus voltage magnitude |V| and angle θ, line current magnitude |I| and angle δ, and even other user-defined quantities, depending on what the PMUs are measuring. The advantage of using (5.1) over (3.1) lies in the flexibility of granting different weights to different network model equations because they have inherently different accuracies due to disparate uncertainties of the model’s parameters. This chapter will focus on constructing nonlinear network models f (x) of AC transmission network, classic HVDC link [86], VSC-HVDC [87], and FACTS devices [88]. Furthermore, different control modes for some components are attempted to be included into the network model for supplying redundant information. Sequentially individual case study is conducted.

5.1

AC transmission network

The physical model of an AC transmission line is the same as that for the linear state estimation. However, the magnitudes and angles of bus voltages and line currents become independent state variables for the nonlinear PMU-based state estimation. Thus the state variable vector x for AC system is ˜ |˜I| θ δ]T xac = [|V| In addition, each network equation in (4.4) has to be expanded into two equations. Hence the network model f (x) for an AC transmission line is formulated as    uac = fac (x) =  

˜ i ||yf i | cos(θi + φf i ) − |V ˜ j ||yf j | cos(θj + φf j ) − |˜If | cos δf |V ˜ ˜ |Vi ||yf i | sin(θi + φf i ) − |Vj ||yf j | sin(θj + φf j ) − |˜If | sin δf ˜ i ||yti | cos(θi + φti ) − |V ˜ j ||ytj | cos(θj + φtj ) − |˜It | cos δt |V ˜ i ||yti | sin(θi + φti ) − |V ˜ j ||ytj | sin(θj + φtj ) − |˜It | sin δt |V

   ,  (5.4)

5.2. Classic HVDC link

43

where

yi + 12 yi0 yi yi 1 ; yf j = ∗ ; yti = ; yf j = yi + yi0 . 2 a a a 2 And θ, δ and φ are the angles of voltage phasor, current phasor and admittance, respectively. The lower part of the measurement model, i.e., v = x − z, is yf i =

vac =

h

˜ − z ˜ | |˜I| − z ˜| θ − zθ δ − zδ |V| |V |I

The Jacobian matrix for the measurement model is  ∂u   ∂fac (x) ∂fac (x) ∂fac (x) ac ˜ ∂θ ∂xac ∂|V| ∂|˜ I| = Hac (x) =   ∂vac Y1 ∂xac

iT

.

∂fac (x) ∂δ

(5.5)

 (5.6)

 

where  ∂fac (x)   = ˜  ∂|V|

|yf i | cos(θi + φf i ) |yf i | sin(θi + φf i ) |yti | cos(θi + φti ) |yti | sin(θi + φti )

or −|yf j | cos(θj + φf j ) or −|yf j | sin(θj + φf j ) or −|ytj | cos(θj + φtj ) or −|ytj | sin(θj + φtj )

   ; 

∂fac (x) = [− cos δ − sin δ − cos δ − sin δ]T ; ∂|˜I|  ˜i ||yf i | sin(θi + φf i ) or |V˜j ||yf j | sin(θj + φf j ) −|V ˜i ||yf i | cos(θi + φf i ) or −|V˜j ||yf j | cos(θj + φf j ) ∂fac (x)   |V = ˜i ||yti | sin(θi + φti ) or ∂θ  −|V |V˜j ||ytj | sin(θj + φtj ) ˜ |Vi ||yti | cos(θi + φti ) or −|V˜j ||ytj | cos(θj + φtj )

   ; 

∂fac (x) = [|˜I| sin δ − |˜I| cos δ |˜I| sin δ − |˜I| cos δ]T . ∂δ Substituting (5.4), (5.5), (5.6) and the predefined weighting matrix W into (5.3), the successive update of x can be calculated.

5.2 5.2.1

Classic HVDC link Network model

It is more realistic to build a nonlinear network model for a classic HVDC link due to its inherent nonlinearity. The proposed nonlinear model also includes the adjunct AC currents as state variables and takes the control modes as a supplement. A simplified classic HVDC link model is shown in Fig. 5.1, where I˜r and I˜i denote the current phasors flowing from the AC side of the rectifier to the DC link and from the DC link to the AC side of the inverter, respectively. Based on the

44

Nonlinear network models 

 



Figure 5.1: A simplified classic HVDC link model [78] available PMU measurements and DC measurements, the network equations can be formulated as [31, 32, 78, 89]: √   |I˜r | − K ∗ 3 π 2 ∗ ar Idc √    Vrdc − K ∗ 3 π 2 ∗ ar |V˜r | cos(θr − δr )  √    Vrdc − K ∗ 3 2 ∗ ar |V˜r | cos α + 3 Xr Idc    π √π   f (x) =  |I˜i | − K ∗ 3 π 2 ∗ ai Idc (5.7) , √    Vidc − K ∗ 3 2 ∗ ai |V˜i | cos(θi − δi )    π √    Vidc − K ∗ 3 π 2 ∗ ai |V˜i | cos δ + π3 Xi Idc  Idc − R1dc (Vrdc − Vidc ) where K = 0.995 is a coefficient for a twelve-pulse AC/DC converter; ar and ai are the tap ratios on the rectifier side and the inverter side, respectively; α and δ 1 (= π − γ 2 ) are the firing angle (also called ignition delay angle) and the extinction delay angle, respectively; Xr and Xi are the transformers’ reactances on the rectifier side and on the inverter side, respectively. |V˜r | and θr are the magnitude and angle of V˜r ; |I˜r | and δr are the magnitude and angle of I˜r , which is the current phasor flowing from the AC side of rectifier to the DC link; the same quantities with subscript i apply to the inverter. For the above seven equations, there are seven DC quantities of concern, which are Vrdc , Vidc , Idc , cos α, cos δ, ar , and ai . Both AC and DC variables in (5.7) will become vectors when multiple classic HVDC links are installed in the system. 5.2.1.1

Classic HVDC control modes

Equation (5.7) characterizes classic HVDC links under steady state without including the control modes. As the referenced values can be maintained by controllers during steady state, it is beneficial to include HVDC control modes into the network model to provide more redundant information. Generally, a classic HVDC link can control two variables at both the rectifier and the inverter: the transformer tap ratio ar and the firing angle α on the rectifier side; the transformer tap ratio ai and the extinction angle γ on the inverter side. The firing 1δ 2γ

here is a DC grid variable, which is different from the angle of an AC line current phasor. is the extinction (advance) angle

5.2. Classic HVDC link

45

angle α keeps a normal operation range within 15◦ to 20◦ with a minimum limit of about 5◦ . The extinction angle γ maintains a minimum limit of 15◦ for 50 Hz and 18◦ for 60 Hz. Controlling the firing/extinction angles is called grid/gate control, and is far more rapid (1 ∼ 10 ms) than tap ratio control (5 ∼ 6 s per step) [78]. Therefore, firing/extinction angle control is used initially for rapid actions, probably followed by tap ratios’ changes to restore the converter quantities, firing and extinction angles, to their normal ranges. Since the slow changes of tap ratios could be easily estimated [23], they are not included into the control mode equations. Herein we assume that ar and ai are maintained at constant values and the firing/extinction angles are used for control purposes. The two most common control modes are briefly discussed below in order to develop the control mode equations, more details can be found in [78, 89]. Rectifier current control mode (RCCM) • When α > αmin , the rectifier maintains constant DC current by changing α. It is the normal constant current (CC) control mode (represented by AB in Fig. 5.2 ). • When α = αmin , the rectifier maintains the constant ignition angle (CIA) control mode (represented by FA). • The inverter always maintains a constant extinction angle (CEA) γ = γ ref control mode (represented by CD). In this control mode, the intersection point E represents the normal operating condition. Inverter current control mode (ICCM) When the rectifier operates at a reduced voltage represented by F’A’B, CD representing the inverter’s CEA operation would not intersect it. Therefore, • The inverter maintains a constant current (represented by GH). • The rectifier maintains a constant firing angle α = αmin (represented by F’A’). The intersection point E’ represents the operating condition at a reduced rectifier voltage. Im in Fig. 5.2 denotes the current margin, which represents the difference between the rectifier current reference and inverter current reference. It is usually set at 10% to 15% of the rated current to ensure that the two constant current characteristics do not cross each other due to errors in measurements or other causes [89].

46

Nonlinear network models

Vdc F

Rectifier (CIA) Normal vol.

C

A

G

E

F’ Reducded vol.

E’

Inverter (CEA)

A’

Inverter (CC)

D

Rectifier (CC)

H

{

B

Idc

Im Figure 5.2: Converter control steady-state characteristics [78, 89] For static state estimation purposes, the control mode equations only consider the equilibrium of each control mode, which are given by:  ref (re) 0 = C1 ∗ (Idc − Idc )     0 = C ∗ (γ − γ ref ) 2 ref (in)  0 = C3 ∗ (Idc − Idc )    0 = C4 ∗ (α − αmin ), ref (re)

ref (in)

(5.8)

where Idc and Idc are the DC current references for the RCCM and ICCM, respectively; C = [C1 C2 C3 C4 ] is the control mode index. Ci = 1 indicates the corresponding equation is activated, otherwise Ci = 0 indicates the corresponding equation will be removed out of the control mode equation set during the state estimation. Hence, C = [1 1 0 0] and C = [0 0 1 1] refer to the RCCM and ICCM, respectively. Replacing γ and α with the state variables cos δ and cos α, respectively, (5.8) is rewritten as:  ref (re) 0 = C1 ∗ (Idc − Idc )     0 = C ∗ (cos δ − cos δ ref ) 2 (5.9) ref (in)  0 = C ∗ (Idc − Idc ) 3    0 = C4 ∗ (cos α − cos αmin ),

5.2. Classic HVDC link 5.2.1.2

47

Interface model between DC Links and AC grids

Since ar and ai are treated as constant parameters, (5.7) only involves five DC states (Vrdc , Vidc , Idc , cos α, and cos δ), and AC voltage phasors V˜r , V˜i and current phasors I˜r , I˜i . However, the current phasors I˜r , I˜i are not state variables of the AC network model as V˜r and V˜i . Hence, Kirchhoff’s current law is applied to the terminal buses that are adjacent to the rectifier or the inverter in order to formulate the current phasors I˜r , I˜i with AC states as Pm Pn (5.10) I˜r = − j=1 I˜rj , I˜i = j=1 I˜ij , where j denotes an AC bus to which bus r (or i) is connected through the AC line rj (or ij); m (or n) denotes the number of buses that are connected to bus r (or i). Consequently, the DC link and the AC grid can be interfaced through the voltage phasors V˜r , V˜i and the current phasors I˜r , I˜i . This DC link and interface models are able to adapt to various topologies of hybrid AC/DC grids. It can represent an embedded DC link in an existing AC grid or as an interconnection between two asynchronous AC grids. Furthermore, it is flexible to be extended to a multi-terminal DC (MTDC) grids [32].

5.2.2

Measurement model for hybrid AC/DC systems

For the hybrid AC/DC state estimation, the AC network model, the classic HVDC link model, as well as their interface model need to be combined together. The state variable vector x for a hybrid AC/DC system is ˜ |˜I| θ δ |˜Ir | |˜Ii | δr δi Vrdc Vidc Idc cos α cos δ]T . x = [|V|

(5.11)

Here voltage and current phasors are applied in polar coordinates, i.e. magnitude and angle. Using phasors in polar coordinates takes two significant advantages: (i) PMU measurements can be directly used without coordinate change; (ii) more importantly, it allows angle bias detection and correction, which will be addressed in Section 5.2.3.3. For a hybrid AC/DC grid, the network model f (x) is formulated as:   fac (x)   f (x) =  fad (x)  , (5.12) fdc (x) where fac (x) is(5.4)), and fad (x) and fdc (x) are formulated as   Pm |˜Ir | cos δr + j=1 |I˜rj | cos δrj   |˜I | sin δ + Pm |I˜ | sin δ  r r rj  j=1 rj fad (x) =  P ,  |˜Ii | cos δi − nj=1 |I˜ij | cos δij  Pn |˜Ii | sin δi − j=1 |I˜ij | sin δij

(5.13)

48

Nonlinear network models          fdc (x) =         

√

2 ˜ K ∗ 3√ π ∗ ar Idc − |Ir | ˜ r | cos(θr − δr ) − Vrdc K ∗ 3 π 2 ∗ a r |V √ 3 2 3 ˜ K∗ √ π ∗ ar |Vr | cos α − π Xr Idc − Vrdc 2 ˜ K ∗ 3√ π ∗ ai Idc − |Ii | 2 ˜ K ∗ 3√ π ∗ ai |Vi | cos(θi − δi ) − Vidc 3 2 ˜ i | cos δ − 3 Xi Idc − Vidc K ∗ π ∗ a i |V π Vrdc − Vidc − Rdc Idc ref(re) ref(in) C1 ∗ (Idc − Idc ) + C3 ∗ (Idc − Idc ) C2 ∗ (cos δ − cos δ ref ) + C4 ∗ (cos α − cos αmin )

         .        

(5.14)

The second part of the measurement model can be expressed as:



   vac xac − zac     v =  vad  =  xad − zad  , vdc xdc − zdc

(5.15)

where vac is (5.5), vad and vdc are shown below

vad = xad − zad =

h 

vdc = xdc − zdc

|˜Ir | − z|˜Ir |

Vrdc − zVrdc   Vidc − zVidc  =  Idc − zIdc   cos α − zcos α cos δ − zcos δ

|˜Ii | − z|˜Ii |     .   

δr − zδr

δi − zδi

iT

,

5.2. Classic HVDC link

49

The Jacobian matrix for a hybrid AC/DC state estimation is :

               H(x) =               

∂fac (x) ˜ ∂|V|

∂fac (x) ∂|˜ I|

∂fac (x) ∂θ

∂fac (x) ∂δ

0 −|˜Ir | sin δr |˜Ir | cos δr

cos δ r sin δ r 0

∂fad (x) ∂|˜ I|

0

∂fad (x) ∂δ

0

−|˜Ii | sin δi |˜Ii | cos δi

cos δ i sin δ i

0

−I ˜ r | sin(θr − δr ) K1 ∗ ar |V ∂fdc (x) ˜ ∂|V|

0

∂fdc (x) ∂θ

∂fdc (x) ∂xdc

−I

0

˜ i | sin(θi − δi ) K1 ∗ ai |V

I 0 0

0 I 0

0 0 I

               ,              

(5.16) where

 ∂fad (x)   =  ∂|˜ I|

cos δrj sin δrj cos δij sin δij

or or or or

0 0 0 0



 ∂fad (x)   = ∂δ 

  , 

  K ∗ a cos(θ − δ )  1 r r r   K ∗ a cos α 1 r    ∂fdc (x)  =  K1 ∗ ai cos(θi − δi ) ˜  ∂|V|  K1 ∗ ai cos δ     

0 or or 0 or or 0 0 0

 0 0

−|˜Irj | sin δrj |˜Irj | cos δrj |˜Iij | sin δij −|˜Iij | cos δij 

             ∂f (x)    dc = 0 ,   ∂θ  0           

or or or or

0 0 0 0

   , 

 0 −K1 ∗ ar sin(θr − δr ) or 0     0   0   −K1 ∗ ai sin(θi − δi ) or 0  ,   0   0    0 0

50

Nonlinear network models 

0 0 K1 ar  −1 0 0    −1 0 −K2 Xr   0 0 K1 ai ∂fdc (x)   =  0 −1 0  ∂xdc  0 −1 −K2 Xi   1 −1 −Rdc    0 0 C1 + C3 0 0 0 √ 3 2 3 K1 = 0.995 ∗ , K2 = . π π

0 0 0 0 K1 ∗ ar |V˜r | 0 0 0 0 0 0 K1 ∗ ai |V˜i | 0 0 0 0 C4 C2

         ,       

The nonlinearities in the Jacobian matrix (5.16) are fewer than those in the Jacobian matrix of a conventional state estimation, which will reduce the computation load. The high degree of sparsity also helps to decrease the computational effort substantially.

5.2.3

Considerations for practical application

In the perfect condition for the PMU-based state estimation, PMUs are installed at all buses so that all the bus voltage and line current phasors are measured. Thus the system is not only fully observable but also of high redundancy. In addition, the measurement noise is too small to be considered. However, in real life this condition can rarely be true, and all the aforementioned issues have to be carefully considered since they may affect the estimation’s performance. Therefore, prior to implementing the proposed nonlinear PMU-based state estimator, all these issues need to be carefully assessed. This section discusses how to analyze system observability with PMUs and the effect of redundancy, what is the allowable measurement noise level of PMUs and DC grid measurements, and how to deal with the phase mismatch owing to imperfect PMUs synchronization. 5.2.3.1

Observability analysis and measurement redundancy

Generally, observability analysis aims to determine whether there are observable islands within the network, and to isolate observable islands. System observability can be analyzed in two main approaches: topological and/or numerical methods. Basic topological methods for conventional state estimations can be found in [16, 90, 91]. A non-iterative numerical method is proposed in [92]. In addition, observability often acts as the criterion for PMU placement in power systems, which aims to maximize system observability with a minimum number of PMUs. If full observability for an entire network is required, the algorithms in [93, 94, 95, 96, 97] may be used for adding new PMUs.

5.2. Classic HVDC link

51

However, observability analysis herein is performed for each individual island, aiming to define whether this island is observable or not as a portion of the power network, and then developing independent state estimator models for each island to be solved. Strictly defining observable islands for an entire system is out of this section’s scope. Since PMUs’ phasor measurements replace power flow and power injection measurements from SCADA, new topological rules for AC systems are defined as follows [94, 98]: • R1: A bus with a PMU installed and any line extending from the bus are observed. • R2: Any bus that is incident to an observed line connected to an observed bus is observed. • R3: Any line joining two observed buses is observed. • R4: If all the lines incident to an observed bus are observed, save one, then all of the lines incident to that bus are observed. • R5: Any bus incident only to observed lines is observed. To define the observability rules for classic HVDC links, it is necessary to account for all the states involved in each link. There are five DC states to be estimated, together with four AC states (|V˜r |, |V˜i |, θr , θi ), and four AC/DC interface states (|I˜r |, |I˜i |, δr , δi ). Observe that for every set of DC link equations, i.e. (5.14), there are 13 states involved. Therefore, the hybrid AC/DC observability algorithm is extended by considering the DC network model: • R6: The DC link is observable if all DC states are measured by metering devices. • R7: If the DC states are not fully measured, at least 4 of 13 states involved in each link model have to be known in order to make all the DC states observable. Note that the AC/DC interface state variables among the four measurements do not have to be measured directly. They can also be accounted as known states by being calculated from AC measurements. However, not all combinations of four measurements will suffice. First, these four measurements should not contain Idc and cos δ for RCCM, or Idc and cosα for ICCM since the control mode equations have already provided references for the above state variables. Second, the four measurements should not all come from only the rectifier side or only the inverter side. In addition, any of the four measured state variables should not be possible to be calculated by the other three measurements. Providing a classic HVDC link under RCCM, (5.14) can be reduced to a five-equation set as

52

Nonlinear network models

follows:

       

˜ r |, θr , δr , Vrdc ) f1 (|V ˜ r |, cos α, Vrdc ) f2 (|V ˜ f3 (|Vi |, θi , δi , Vidc ) ˜ i |, Vidc ) f4 (|V f5 (Vrdc , Vidc ).

    .   

(5.17)

Based on (5.17), the combinations can be divided in two types: two measurements from the rectifier side and two from the inverter side, three measurements from the rectifier side and one from the inverter side. For the first type, all the combinations can be inferred by the following steps: ˜ i | or Vidc . • S1: The first measurement selected from inverter side can be |V Knowing either of them can calculate the other one by using f4 . Moreover, Vrdc will be known sequentially by f5 . • S2: Knowing either θi or δi can calculate the other. So far, all the states from the inverter side have been known. • S3: Since Vrdc has been calculated, there are two equation f1 and f2 for four ˜ r |, θr , δr , and cos α. Therefore, any two of the unknown states, which are |V ˜ r | and cos α simultaneously. four states can be selected, except for choosing|V • S4: In total, the number of combinations is 2 × 2 × (C(4, 2) − 1) = 20. For the second type, the steps are as follows: • S1: Select any three measurements among five states on the rectifier side except ˜ r |, cos α, Vrdc together. Hence, all the states on the rectifier for selecting |V side can be calculated associated with f1 and f2 . ˜ i | is known by f4 . • S2: Vidc will be sequentially known by f5 and then |V • S3: Either θi or δi can calculate the other. So far, all the states from the inverter side have been known. • S4: In total, the number of combinations is (C(5, 3) − 1) × 2 = 18. All the proper combinations of four measurements to make a classic HVDC link observable are shown in Table 5.1. The upper part of Table 5.1 presents the first type of combinations, whereas, the lower part is for the second type of combinations. As PMUs measure synthetic magnitude and angle in phasor coordinates, in reality extra measurements out of each combination might be obtained incidentally. For ˜ i |, and θi ; |V ˜ r |, and |˜Ir | would be instance, when using the combination of θr , δr , |V known automatically when θr , δr are measured or calculated. As indicated by the red marks in Table 5.1, in the case of no measurement redundancy, the DC states can still be computed even without any DC measurement.

5.2. Classic HVDC link

53

Table 5.1: Proper combinations of four measurements to make a classic HVDC link observable |V˜r |

θr

δr

∗

∗ ∗ ∗

∗

∗ ∗

∗ ∗ ∗

∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗ ∗

∗

θi

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

δi

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗

Vidc

∗ ∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

|V˜i |

∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

cosα

∗ ∗ ∗

∗ ∗

Vrdc

∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

54

Nonlinear network models

This brings a great advantage since AC measurements are readily provided by PMUs. When measurements are redundant, this can be taken as an advantage for cross-validation [99], as to be discussed in Section 5.2.3.3. In addition, the tap ratios of the rectifier and inverter can also be included as DC states when needed as there are four more equations than the number of DC states. A complementary approach is to examine the numerical properties of the Jacobian matrix of the hybrid AC/DC model, which is referred to as the numerical method. rank((H)T (H)) = Nx ,

(5.18)

where H(x) is the Jacobian matrix; and Nx represents the number of state variables. Generally, the more rows that the Jacobian matrix has w.r.t. the number of columns, the higher redundancy offered by the PMU-based state estimation. As mentioned in [16, 23, 90], measurement redundancy is crucial for bad data detection and identification. When a measurement is lost, its corresponding rows in the measurement model equation, in the Jacobian matrix, and in the weighting matrix will be removed. As long as (5.18) holds, all the states can be estimated by using the nonlinear WLS algorithm. A case study for a low measurement redundancy scenario is presented in Section 5.2.4.3 to investigate its effect on the PMU-based state estimation. 5.2.3.2

Measurement noise and the choice of weightings

In reality, it is inevitable to have noise in measurements due to (i) the instrument transformers, (ii) the cables connecting the instrument transformers to the sensors or A/D converters, and (iii) the sensors or A/D converter [90]. The standard uncertainty (σ) for each measurement is proportional to the specified maximum uncertainty (σmax ) of the PMU with a coefficient of √13 [100]. Nevertheless σmax varies for different PMU vendors, and Table 5.2 gives an example [101]. It is assumed that the DC measurements have a σmax of 0.01% of reading or 0.001% range. Then the σmax values are transfered into signal-to-noise ratios (SNRs) using (5.19) in order to add Gaussian noise to the true values.  √ SN R = 10 ∗ log10

3

2

dB. (5.19) σmax As discussed in Chapter 3, when the measurement has a Gaussian distribution, the WLS solution is equivalent to the MLE if the inverse of each measurement Table 5.2: σmax for different variables from one PMU vendor |V | |I| θ and δ

0.02% of reading or 0.002% range 0.03% of reading or 0.003% range 0.01◦ or 10% of range minimum

5.2. Classic HVDC link

55

Table 5.3: SNRs and weights for different measurements Meas. |V | |I| θ and δ DC states

SNR 78.75 dB 75.23 dB 79.93 dB 84.77 dB

Weighting 7.5 ∗ 107 3.3 ∗ 107 9.8 ∗ 107 3 ∗ 108

covariance is used as its weight. Therefore, the SNRs and the weights of different measurements are calculated and shown in Table 5.3. The weightings for the network model equations depend on the modeling accuracy for each component. A case study on measurement noise and weighting selection is presented in Section 5.2.4.4, where the weights for the network model equations are all equal to the highest measurement weight. 5.2.3.3

Angle bias detection and correction by PMU-based state estimation

Angle biases (or shifts) emerge due to imperfect synchronization or incorrect timetagging by PMUs [22, 102, 103]. These phase angle errors have been observed from recorded data in several utilities [22, 23]. Reference [104] presents two time skew cases that result in angle biases. In one case, the GPS signal cable was loosely connected so the signal was intermittent. Thus, the PMU time was not accurate, resulting in spikes on top of the correct angle values. The other case of angle bias occurred due to drifting of the internal clock. Different from measurement errors, angle bias does not have a normal distribution and its deviation varies within 1 ∼ 2 degrees, even 20 degrees in some extreme cases. In addition, angle bias could last for a few snapshots, which may contaminate other measurements. Since magnitude and angle are independent states herein, angle biases can be detected by using redundant measurements [22]. The vector of angle bias variables Ω is included in the state vector as ˜ |˜ x = [|V| I| θ δ |˜ Ir | |˜ Ii | δr δi Vrdc Vidc Idc cos α cos δ Ω]T where Ω = [Ωθ Ωδ Ωr Ωi ] The voltage and current angles in the measurement error vector become θ − zθ + Ωθ and δ − zδ + Ωδ Similarly, the angles of the AC/DC interface currents in the measurement error vector become δr − zδr + Ωr and δi − zδi + Ωi

56

Nonlinear network models

In order to correct the angle bias, it is required that rank((H)T (H)) = Nx + NΩ ,

(5.20)

where NΩ represents the number of angle bias auxiliary state variables. This approach has been defined more formally in the control literature, and it has been termed as cross-validation. Mathematical proofs for cross-validation rules can be referred to [99]. Angle bias variable Ω greatly facilitates angle bias correction. Compared to common bad data detection and correction methods, such as using normalized residuals [90], it does not need to perform additional calculations or to define a threshold to determine bad data. In addition, it avoids the risk that the largest normalized residual method may fail in the detection of gross errors for the measurements that have a large undetectable component [105]. In fact, this angle bias correction gives a huge flexibility for correcting angle bias no matter how large the bias is. The requirement, however, is to have enough redundancy to accommodate the angle bias variables Ω. Hence, if the power system has high measurement redundancy, this approach would be suitable. Reference [102] applies the same algorithm for magnitude error detection and correction. However, the authors believe it is not necessary to use this method to correct magnitude errors, because the incorrect time tagging issues normally do not give rise to deviations in the magnitude measurements. It is not worth incorporating measurements redundancy to detect and correct minor magnitude errors. A case study for the angle bias scenario is presented in Section 5.2.4.5.

5.2.4

Case study

The 6-bus system and the KTH-Nordic 32 system are modified for this case study. In the 6-bus system a classic HVDC link is established between bus 3 and bus 4. In the KTH-Nordic 32 system a classic HVDC link replaces the original AC line between bus 36 and bus 41. The first two subsections present estimation results when all the weightings for both network equations and measurements are assumed to be 1, and full measurement observability is assumed. The following three subsections present simulation results for a scenario with low measurement redundancy, a scenario with measurement noises and corresponding weights, and a scenario where angle bias correction is performed. Finally, simulation time and computation performance are discussed. 5.2.4.1

NSE for the 6-bus hybrid AC/DC system

A circuit breaker located on line 4 between bus 4 and 6 was opened at t = 5 s and after three cycles it was re-closed at t = 5.06 s. The DC link was under the normal ref operation condition with Idc = 0.506 p.u. and cos δ ref = 0.951 p.u. for RCCM.

5.2. Classic HVDC link

Vmag−true

57

Vmag−meas.

Vang−true

Vmag−est.

1

0.95 1

Vang−est.

2

3

4

5

10 5 0 1

6

Bus No.

2

3

4

5

6

5

6

Bus No.

Vang−estimation−residual

−15

Vmag−estimation−residual

−16

x 10 Error (deg)

x 10 Error (p.u.)

Vang−meas.

15 θ(deg)

|V|(p.u.)

1.05

2 1

5 0 −5

0 1

2

3

4

5

6

1

Bus No.

Iri−true

Iri−meas.

2

3

4 Bus No.

˜ at all buses (a) |V|

(b) θ at all buses (bus 2 is the reference) Hvdc−true

Iri−est.

Hvdc−meas.

Hvdc−est.

1

0.5

p.u.

p.u.

1

0

0.6

−0.5 1 |Ir|

1.5

2 |Ii|

3 δ r

3.5

4 δi

1 Vrdc

1.5

2 Vide

x 10 Error (p.u.)

4 2

2.5

3 Idc

3.5

4 cosa

4.5

5 cosg

4.5

5 cosg

Hvdc−estimation−residual

−16

Iri−estimation−residual

−16

x 10 Error (p.u.)

2.5

6

0 1 |Ir|

0.8

8 7 6 5

1.5

2 |Ii|

2.5

3 δ r

3.5

(c) AC/DC interface states

4 δ i

1 Vrdc

1.5

2 Vide

2.5

3 Idc

3.5

4 cosa

(d) DC states

Figure 5.3: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link for a single snapshot And the state vector for the 6-bus hybrid AC/DC system is defined as follows x = [|Ve1 |, |Ve2 | . . . |Ve6 |, θ1 , θ2 , . . . θ6 , |Ie1 |, |Ie2 | . . . |Ie12 |, δ1 , δ2 , . . . δ12 , |Ier |, |Iei |, δr , δi , Vrdc , Vidc , Idc , cos α, cos δ]T . Figure 5.3 shows the state estimation results for one single snapshot, and Fig. 5.4 is for multiple snapshots. As the network model is relatively similar to the model in PSAT, the residuals for one single snapshot are extremely small, lower than 10−14 p.u. However, when the system was subject to a perturbation, the estimation residuals increased as shown in Fig. 5.4. This is due to two reasons:

58

Nonlinear network models Vmag−true

Vmag−meas.

Vmag−est.

1

Vang−true

θ(deg)

|V|(p.u.).

0.95

0.9 0

2

4

6

8

12 10 time (s)

14

Vang−est.

16

4 2 0

18

2

4

6

8

10 12 time (s)

14

16

18

0

Vmag−residual−error

10

Error (deg)

0

10 Error (p.u.)

Vang−meas.

6

Vang−residual−error

−10

10

−10

10

−20

10

−20

10

0

2

4

6

8

10 12 time (s)

14

16

Vidc−meas.

4

6

8

cosα−true

Vidc−est.

cosα (p.u.)

1.2 Vidc (p.u.)

2

10 12 time (s)

14

16

18

(b) θ at bus 4 (bus 2 is the reference)

(a) |V˜ | at bus 4 Vidc−true

0

18

1.15

cosα−meas.

cosα−est.

1 0.98 0.96 0.94

1.1 0

2

4

6

8

10 12 time (s)

14

16

2

18

Vidc−residual−error

0

8

10 12 time (s)

14

16

18

16

18

cosα−residual−error

0

Error (p.u.)

Error (p.u.)

6

10

10

−10

10

0

−10

10

−20

−20

10

4

10 2

4

6

8

10 12 time (s)

14

(c) DC Voltage at bus 4

16

18

0

2

4

6

8

10 12 time (s)

14

(d) cos α

Figure 5.4: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link for multiple snapshots • Lack of model and/or topology update during the perturbation. A circuit breaker was opened and re-closed after three cycles. During this period, the network model was not updated accordingly, resulting in a large estimation residual. • The limitation of static network models. During the transient dynamic period after the perturbation, the controllers and/or the components that have dynamical properties respond to the changes. For the AC network model, the Kirchhoff’s circuit law is still valid given correct admittance matrix. (AC lines with FACTS devices can be a different case). However, for the static classic HVDC link model, as some of its states are directly controlled or indirectly affected by the controllers, it may not hold during transient dynamics. This indeed explains in Fig. 5.4 that the performance of the state estimation gradually came back to a normal level after the drop, but still had

5.2. Classic HVDC link

59

larger residuals compared to that in steady state. Nevertheless, the estimation residuals are within an acceptable error range. 5.2.4.2

NSE for the KTH-Nordic 32 hybrid AC/DC system

Herein a 400 kV classic HVDC link is established between bus 36 and bus 41. A 0.4 p.u. (10%) load increase was applied at bus 41 at t = 5 s. And the state vector for the KTH-Nordic 32 hybrid AC/DC system is organized as follows x = [|V˜1 |, |V˜2 | . . . |V˜53 |, θ1 , θ2 , . . . θ53 , |I˜1 |, |I˜2 | . . . |I˜160 |, δ1 , δ2 , . . . δ160 , |I˜r |, |I˜i |, δr , δi , Vrdc , Vidc , Idc , cos α, cos δ]T . Vmag−true

Vmag−meas.

Vmag−est.

Vang−true

θ(deg)

|V|(p.u.)

1.1 1 0.9

10

|Error (p.u.)

30 Bus No.

40

−100

50

10

x 10

10 5 10

20

30 Bus No.

40

Iri−meas.

30 Bus No.

40

50

Vang−estimation−residual

x 10 2 0 −2 −4

10

50

20

30 Bus No.

40

50

(b) θ (bus 9 is the reference)

(a) |V˜ | at all buses Iri−true

20

−13

Vmag−estimation−residual

−15

15

20

Vang−est.

−50

Error (deg)

0.8

Vang−meas.

0

Iri−est.

Hvdc−true

2

Hvdc−meas.

Hvdc−est.

1 p.u.

p.u.

1

0

0.96

−1 |Ir|

|Ii|

δi

Vrdc

2

1 0.5 δr

(c) AC/DC interface states

δi

Idc

cosa

cosg

Hvdc−estimation−residual

−14

1.5

|Ii|

Vide

x 10 Error (p.u.)

Error (p.u.)

δr

Iri−estimation−residual

−13

x 10

|Ir|

0.98

15 10 5 Vrdc

Vide

Idc

cosa

cosg

(d) DC states

Figure 5.5: NSE for the KTH-Nordic 32 hybrid AC/DC with a classic HVDC link for a single snapshot

60

Nonlinear network models Vmag−true

Vmag−meas.

Vmag−est.

0.852

0.848 0

2

4

6

8 10 time (s)

12

14

16

10

Error (deg)

Error (p.u.)

−67

−67.5 0

10

−10

10

10 0

2

4

6

8 10 time (s)

12

14

6

Vidc−true

Vidc−meas.

cosα (p.u.) 6

8 10 time (s)

14

16

−20

0

2

4

6

8 10 time (s)

12

12

14

cosα−meas.

cosα−est.

0.966 0.9655 0.965 0.9645 2

16

4

6

8 10 time (s)

12

14

16

cosα−residual−error

Vidc−residual−error

0

16

−10

cosα−true

0.942 4

14

Vang−residual−error

Vidc−est.

0.944

2

12

(b) θ at bus 41 (bus 9 is the reference)

0.946

0.94 0

8 10 time (s)

16

(a) |V˜ | at bus 41

Vidc (p.u.)

4

0

−20

10

0

10 Error (p.u.)

Error (p.u.)

2

Vmag−residual−error

0

10

−10

10

−20

−20

10

Vang−est

−66.5

0.85

10

Vang−m

−66 θ(deg)

|V|(p.u.)

Vang−true

0.854

0

2

4

6

8 10 time (s)

12

(c) DC Voltage at bus 41

14

16

10

0

2

4

6

8 10 time (s)

12

14

16

(d) cos α

Figure 5.6: NSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC link for multiple snapshots As shown in Fig. 5.5, the estimation residuals are comparable to results in Section 5.2.4.1, which indicates that the accuracy of the proposed state estimator is not affected by the size of the test system. As this case study has no topology change during the perturbation, the estimation residual in Fig. 5.6 does not experience a leap as in Fig. 5.4. The increase of the estimation residual after the instance when the perturbation occurred is due to lack of representation of the system’s dynamic behavior. Figure 5.6 shows that the estimation residuals are within an acceptable error range even when the system is subject to a dynamic change. 5.2.4.3

NSE for the scenario without DC measurements

The same test scenario as in Section 5.2.4.1 was applied here. To decrease the measurement redundancy, only the voltage phasor measurements at bus 3 and 4,

5.2. Classic HVDC link Vmag−true

61

Vmag−meas.

Hvdc−true

Vmag−est.

Hvdc−meas.

1.05

p.u.

|V|(p.u.)

1 1

0.8 0.6

0.95 1

2

3

4

5

6

Vrdc

Bus No.

Vmag−estimation−residual

−16

x 10

cosa

Idc

cosg

Hvdc−estimation−residual

−15

x 10 3 Error (p.u.)

2 Error (p.u.)

Vide

1

0 1

2

3

4

5

6

2.5 2 1.5 Vrdc

Bus No.

(a) |V˜ | at all buses for one snapshot Vmag−true

Vmag−meas.

Vide

Idc

cosg

cosa

(b) DC states for one snapshot

Vmag−est. V

1.02

−true

idc

V

−meas.

idc

V

−est.

idc

1

Vidc (p.u.)

|V|(p.u.).

1.2

0.98 0.96 0

2

4

6

8 10 time (s)

12

14

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1.1 0

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6

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16

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0

Error (p.u.)

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10

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10

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10

−10

10

−20

10

−20

10

8 10 time (s)

2

4

6

8 10 time (s)

(c) |V˜ | at bus 5

12

14

16

0

2

4

6

8 10 time (s)

(d) DC Voltage at bus 4

Figure 5.7: Effect of reducing AC/DC measurements redundancy: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link

and current phasor measurements on lines 3, 4, 7, and 8, which are incident to either bus 3 or 4, were provided. There was no DC state measurement provided and the estimation results are shown in Fig. 5.7. Comparing the results in Fig. 5.7 with that in Section 5.2.4.1, reducing measurements redundancy does not significantly influence the estimation performance as long as the measurements can satisfy the observability requirements discussed in Section 5.2.3.1. This test scenario illustrates that when PMUs are available only at critical boundary buses between the AC system and the DC link, it is possible to estimate DC states without having any DC measurements.

62

Nonlinear network models Vmag−true

Vmag−meas.

Vmag−est.

Hvdc−true

Hvdc−meas.

Hvdc−est.

1.05

p.u.

|V|(p.u.)

1 1

0.8 0.6

0.95 1

2

3

4

5

6

−6

Vmag−estimation−residual

x 10 Error (p.u.)

Error (p.u.)

x 10 7 6 5 4 3 1

2

3

4 Bus No.

(a) |V˜ | at all buses

5

6

Vide

Vrdc

Bus No.

Idc

cosa

cosg

Hvdc−estimation−residual

−5

3 2 1 Vrdc

Vide

Idc

cosa

cosg

(b) DC states

Figure 5.8: Effect of measurement noises: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link for a single snapshot 5.2.4.4

Results for the scenario with measurement noises and corresponding weights

The effect of measurement noises and weights selection were studied using the same test scenario as in Section 5.2.4.1 by adding Gaussian white noise and using the weights in Table 5.3. Figure 5.8 shows the estimation results, in which the residuals are larger in comparison with the previous case in Section 5.2.4.1. Nevertheless, the state estimates are acceptable for the applied signal-to-noise ratios. 5.2.4.5

Results for the scenario with angle bias correction

An example of angle bias correction was made for the 6-bus hybrid AC/DC system, where a 7.5◦ angle jump at bus 1 and 30◦ angle jump on line 1, which is incident to the bus 1, were applied at t = 10 s and removed at t = 11 s as shown in Fig. 5.9. This test scenario shows that the proposed state estimator also has the ability of correcting angle biases for hybrid AC/DC grids. 5.2.4.6

Comparison between the linear and nonlinear SEs

Chapter 4 presents a linear state estimation, while this chapter proposes a nonlinear one. A linear, or nonlinear WLS is determined by the measurement error vector e(x). In most cases, this vector presents the same linearity (or nonlinearity) as the network model. The rationale to use the nonlinear state estimation is explained below. For the AC part, both algorithms use the same linear network model. However, since the state variables used in this chapter are in polar coordinates, each linear equation needs to be rewritten into two equations associated with trigonometric

5.2. Classic HVDC link

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63

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100

δ (deg)

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20 10 0 1

2

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6

Bus No.

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−15

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Error (deg)

x 10

2

5 0 −5 1

2

3

4

5

10

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10

12

Iang−estimation−residual

4 2

2

4

6 Line No.

8

(b) Current angles for one snapshot

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Iang−true

40 θ (deg)

−14

8

0

6

(a) Voltage angles for one snapshot (bus 2 is the reference Vang−meas.

6 Line No.

6

Bus No.

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4

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Iang−est.

20 δ (deg)

30 20

0 −20 −40

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10 12 time (s)

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0

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−10

10

−20

10

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10

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8

2

4

6

8

10 12 time (s)

14

16

18

0

2

4

6

8

10 12 time (s)

(c) Voltage angle at bus 1 (bus 2 is the refer- (d) Current angle on line 1 (between bus 1 and bus 3) ence)

Figure 5.9: Angle bias correction: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link

64

Nonlinear network models

Vm−mag

Vang−true

Vmag−est

10

1.04

5

θ (deg)

|V| (p.u.)

Vmag−true 1.06

1.02

Vm−ang

0 −5

1 0.98 0

2

4

6

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−3

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0 0

6 Bus No.

Error (deg)

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Vang−est

8

2 0 −2 0

10

2

4

Bus No.

6 Bus No.

(a) Voltage magnitudes at all buses

(b) Voltage angles at all buses

Figure 5.10: LSE for the 9-bus test system when a 7.5◦ angle bias was introduced to bus 8 for a single snapshot Vmag−m

Vang−true

Vmag−est 10

1.04

5

θ(deg)

|V|(p.u.)

Vmag−true 1.06

1.02 1

Vang−m

0 −5

0.98 0

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4

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8

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−16

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2 1 0 0

6

8

10

8

10

Bus No.

Error (deg)

Error (p.u.)

3

Vang−est

x 10

Vang−residual−error

−14

0.5 0

−0.5 2

4

6

8

10

Bus No.

(a) Voltage magnitudes at all buses

0

2

4

6 Bus No.

(b) Voltage angles at all buses

Figure 5.11: NSE for the 9-bus test system when a 7.5◦ angle bias was introduced to bus 8 for a single snapshot functions of the phasor angles. These trigonometric functions introduce nonlinearities. Although this brings additional computational burden, using phasors in polar coordinates gives the significant advantage of allowing angle bias detection and correction. To support this point, a comparison was performed on the same 9-bus AC test system as in Chapter 4 by using the linear and nonlinear state estimators, respectively. A 7.5◦ angle bias was introduced at bus 8 and 30◦ angle biases at the

5.2. Classic HVDC link

Hvdc−true

Hvdcm

65

Hvdc−est

Hvdc−true

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p.u.

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0.8 0.6

0

Vide

Error (p.u.)

3

x 10

−10

Idc

Vrdc

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Hvdc−residual−error

Idc

cosa

cosg

Hvdc−estimation−residual

Error (p.u.)

Vrdc

2 1

x 10

−16

8 7 6 5

0

Vrdc

Vide

Idc

Vrdc

Vide

Idc

cosa

cosg

(a) DC states in linear SE for one snapshot (b) DC states in nonlinear SE for one snapshot

Figure 5.12: Comparison between the linear (left) and nonlinear (right) classic HVDC link models for LSE and NSE, respectively

lines that are connected with bus 8 for both cases. Figure 5.10 shows the linear state estimation results for the voltage magnitudes and angles when angle biases were introduced. The estimation accuracy for both magnitudes and angles reduces significantly compared to the results presented in Chapter 4 when no angle bias was introduced: magnitude residuals from 10−16 to 10−3 ; angle residuals from 10−15 to 1 deg. This indicates that the angle biases were not successfully detected; moreover, they contaminated other angles and even magnitudes being estimated. Therefore, it is verified that the linear state estimator is unable to detect and correct angle biases. Reference [106] also shows a simple example that draws the same conclusion. On the contrary, the nonlinear state estimator presented in this chapter successfully detected and corrected the angle bias. As shown in Fig. 5.11, estimation accuracy is similar to the case when no angle bias was introduced. The limitations of linear state estimation was reported in [106], even in the case where there is full observability and redundant measurements (PMUs installed at all buses and measuring all the currents at each bus), the linear state estimator fails to provide the correct measurement residuals in the presence of an angle bias. The DC network model can be simplified to a linear one as presented in Chapter 4 by reducing the number of equations and using complex variables. However, this will lead to the following consequence: the state variables that must be used, for instance |V˜r | cos α, are not consistent with other variables, and this may lead to matrix conditioning issues during the linear least squares solution. A comparison between the linear classic HVDC link model as in Chapter 4 and the nonlinear one was carried out and the results are shown in Fig 5.12. Compared

66

Nonlinear network models Table 5.4: Statistics of the computation time Computation time

No. of meas. sets (1000 in total )

%

t ≥ 0.02s 0.01 ≤ t < 0.02s 0.006 ≤ t < 0.01s 0.003 ≤ t < 0.006s t ≤ 0.003s

4 5 113 641 237

0.4% 0.5% 11.3% 64.1% 23.7%

to the nonlinear model’s estimation residual of 10−16 , the linear one only achieves 10−10 . 5.2.4.7

Simulation time and computation performance

Referring to the estimation results in Section 5.2.4.1 and 5.2.4.2, it is observed that most of the estimation errors are below 10−12 with relatively few iterations. Residuals that are higher than 10−12 can result from two reasons: (i) the limitation of the static network models, which can hardly represent the system’s behavior during transient dynamics; (ii) lack of model and/or topology update during the perturbations. For instance, in the test scenario of Section 5.2.4.1, the number of iterations for each estimation snapshot remained around 2 when the system was in steady state. During the first three cycles after the perturbation occurred, the iteration number was 37 at once, and then decreased to 4 after another three cycles. The computation time statistics for the test scenario in Section 5.2.4.1 are shown in Table 5.4. Two hundred and fifty out of 1000 measurement sets were obtained before the perturbation; the others were after the perturbation. The measurement sets for which the estimation computation time was below 0.003 s account for 23.7% of the whole measurement sets; more than 99.6% of the estimation computations are faster than the measurement rate, which is 0.02 s. Computation time exceeding the measurements interval mainly occurred during the perturbation period. The time-performance of the algorithm in a standard PC (2.80 GHz Intel Core processor running matlab R2012b) is already acceptable for real-time applications with minimum delay, and could be improved if the code is optimized and re-implemented in a low-level programming language such as C++. In addition, it demonstrates the feasibility of implementing the estimation algorithm into a real-time application.

5.3

VSC-HVDC

VSC-based HVDC systems have been successfully deployed in various projects globally. Compared to the classic HVDC technology, VSC-HVDC technology offers a key advantage of independent control of active and reactive powers, together with additional benefits in control flexibility and reliability [25].

5.3. VSC-HVDC

5.3.1

67

VSC substation model

To match the PMU measurements in AC systems, DC measurement units (DMUs) could be easily implemented by extending the functionalities of the acquisition system used in converter stations to gather the DC measurements. Based on the knowledge that switching frequency of transistors in VSCs is high and so is the control response, DMUs should be capable to provide high speed and GPS timesynchronized measurements, at least not slower than PMUs. A basic diagram of a VSC substation is shown in Fig. 5.13. It comprises an AC/DC converter, DC capacitor(s), a phase reactor, a transformer and filter(s). The converter requires self-commutating switches, such as insulated-gate bipolar transistors (IGBT), which have a turn-on and turn-off capability. The DC capacitor on the DC side maintains the DC voltage across the VSC. However, in reality, DC voltage may contain ripples due to the harmonic currents in the DC circuit generated during VSC operation. The interface transformer and phase reactors are installed to reduce the fault current and the harmonic current content in the AC current, and enable the control of active and reactive power flow. The AC filter aims to filtering the high-order harmonics introduced by the commutation valve switching process [25]. In Fig. 5.13, V˜ , I˜ are the voltage and current phasors at Bus i, which is also the point of common coupling (PCC); V˜f is the voltage phasor at the AC filter and I˜f is the current phasor flowing into it; V˜v and I˜v are the voltage and current phasors at the AC side of the converter; Zt and Zp are the impedances of the transformer and phase reactor, respectively.

I dc

V I

Zt

PCC Bus i

a :1

Vf If AC filter

Zp

Vv

Phase reactor

Iv



Vdc



Figure 5.13: Basic diagram of a VSC substation As indicated in Fig. 5.13, Bus i connects to the VSC substation through an AC line, which can be represented by the AC branch model to build up the relation ˜ V˜f , I˜f and V˜v , I˜v . Hence, the static network model is formulated as: among V˜ , I, 

 V˜v − V˜f − I˜v Zp  ˜ t  f (x) =  V˜f − a1 V˜ − aIZ . ˜ ˜ ˜ Iv − If − aI PCC Bus i

If AC filter

Phase reactor

I dci 

I dcj

…...

(5.21) 

Vdcj

Vdci





…...

Phase reactor

AC filter

68

Nonlinear network models

Furthermore, in order to find the relation between converter’s voltage and current phasors with the DC voltage and current, the converter model needs to be included. For state estimation purposes, an average value model (AVM) is sufficient for a VSC, which avoids distinguishing different switching levels and modulation types, and instead it focuses on the voltage and current components of the fundamental frequency. An AVM can be represented by a combination of controllable three-phase AC voltage sources connected to the AC circuit and a controllable current source connected to the DC circuit. These three phase voltages are controlled by their respective modulation indexes, namely: ma , mb and mc . Therefore, the relationship between the AC and DC circuits can be formulated as:  a   vv = ma Vdc (5.22) vvb = mb Vdc   c vv = mc Vdc . Besides, regardless of the power flow direction, the power consumption on one side must be always equal to the power injection from the other side. Thus, Idc =

vva iav + vvb ibv + vvc icv = ma iav + mb ibv + mc icv . Vdc

(5.23)

All the AC variables in (5.22) and (5.23) use lowercase letters, which indicates they are instant variables; Vdc and Idc use uppercase letters because when using the AVM, DC variables are relatively constant without considering the harmonics. Then the converter model is formulated as " # Mv Vdc − |V˜v | f (x) : . (5.24) (Kd2a )2 Vdc Idc − |V˜v ||I˜v | cos(θv − δv ) where Mv is the modulation index, which is defined here as the ratio of the rootmean-square (RMS) value of the modulating wave, i.e., positive-sequence component of the AC voltage at the converter, to the peak value of the carrier wave, i.e., the pole-to-pole DC voltage; Kd2a is the coefficient that transfers DC quantities to the AC base when using per unit values; θv and δv are the angles of the converter’s voltage and current, respectively. The first equation uses the modulation index to build the relation between the converter’s AC side voltage and DC voltage; while the second equation models the active power equality at the converter’s AC side and DC side when omitting its internal losses. In summary, combining (5.21) and (5.24) yields a VSC substation model for static state estimations. The state vector of an AC system with VSC substations is given by   ˜ |˜I|, θ, δ, |V ˜ f |, θf , |˜If |, δf , |V ˜ v |, θv , Vdc , Idc T . x = |V|, (5.25)

5.3. VSC-HVDC

5.3.2

69

VSC control modes

Generally, VSCs contain a two-level control strategy. The high-level control provides accurate voltage references for AVM (or for each sub-module of a detailed model), so as to maintain the control variables within an acceptable range. Among the high-level control scheme, vector-current control has been successfully applied on several VSC-HVDC link installations. Inside vector-current control, it has a two-loop control scheme, referred to as the outer control loop and the inner control loop. The outer control loop transfers ref ref the VSC control references, i.e. P ref , Qref , Vdc or Vac , into converter current ref ref T ref references in dq coordinate as idq = [id iq ] . The inner control loop transfers ref ref T the converter current references iref iq ] into converter bridge voltage as dq = [id ref vdq = [vdref vqref ]T . Then, they are transformed to three-phase quantities, which are the converter voltage references in three phases. Finally, a low-level control scheme is utilized, such as pulse width modulation (PWM) algorithm, to regulate the switching signal generation via the voltage references generated by the high-level control, and further to provide switching pulses for valves. This is out of the scope of this application because the converter of an AVM is represented by a voltage source, where no switching device is considered. Depending on the operation mode, reference iref can be determined from either d the active power reference or the DC voltage reference. Similarly, reference iref q can be determined from either the reactive power reference or the AC voltage reference. For each VSC substation, only one iref and only one iref are allowed to q d be implemented. Thus, a choice of d control reference (choosing between Pref and ref ref Vdc ), and q control reference (choosing between Qref and Vac ) has to be made. Therefore, control modes for a VSC substation are formulated as follows: ( ref 0 = CP ∗ (|Vei | ∗ |Iei | ∗ cos(θi − δi ) − P ref ) + CV d ∗ (Vdci − Vdci ) (5.26) ref 0 = CQ ∗ (|Vei | ∗ |Iei | ∗ sin(θi − δi ) − Qref ) + CV a ∗ (|V˜i | − Vi ), where C = [CP CV d CQ CV a ] is the control model index. Ci = 1 indicates the corresponding control mode is activated, otherwise Ci = 0 indicates the corresponding control mode is disabled. Note that to include the control modes into the state estimator, only the equilibrium condition of a system is taken into account, which means the controlled variables are presumed to be equal to the references. There are two reasons to make this assumption. First, the network model for the static state estimations does not include differential equations to represent system dynamics. Nevertheless, fast rate of the PMU-based state estimation will compensate for it as the Kirchhoff’s laws of the network model should hold at each measurement snapshot. Second, vector-current control is rather fast and robust, particularly, it is facilitated with many saturation limiters to protect hardware from being damaged by peak currents after perturbations occur. Hence, the discrepancy of controlled variables is within a small range even during a perturbation, which would not greatly detract the

70

Nonlinear network models

accuracy of the proposed control modes equations. Therefore, including control modes into the state estimator improves its robustness and redundancy. In summary, by combining (5.21), (5.24) and (5.26), a VSC substation model with different control modes can be formulated as:  ˜  Vv − V˜f − I˜v Zp  V˜ − 1 V˜ − aIZ  ˜ t  f a   ˜    Iv − I˜f − aI˜    . (5.27)  f (x) =  Mv Vdc − |V˜v |      (Kd2a )2 Vdc Idc − |V˜v ||I˜v | cos(θv − δv )   ref  CP ∗ (|V˜f | ∗ |I˜f | ∗ cos(θf − δf ) − P ref ) + CV d ∗ (Vdc − Vdc )  CQ ∗ (|V˜f | ∗ |I˜f | ∗ sin(θf − δf ) − Qref ) + CV a ∗ (|V˜f | − V ref ) f

5.3.3

Point-to-point VSC-HVDC link model I dc

V  A single VSC may operate as a FACTS device, such as the static synchronous Z I Z Vf V Vdc compensator (STATCOM) and the static synchronous series compensator (SSSC),  PCC I I Phase which will in next section. A more common application of VSCs is the a :1 be introduced reactor AC Bus i filterVSC-HVDC link, as shown in Fig. 5.14. Two VSC substations are point-to-point connected through a long distance DC cable. Since VSC is a bidirectional converter, it enables to change the power flow direction. One VSC acts the rectifier and the other one will act as an inverter, depending on the power flow control references and settings. t

p

f

v

v

I dci 

…...

Vdcj

Vdci



PCC Bus i

AC filter

Phase reactor

I dcj 



…...

Phase reactor

PCC AC filter

Bus j

Figure 5.14: A point-to-point VSC-HVDC link model For the point-to-point VSC-HVDC link, each converter can be modeled as a VSC substation as in (5.27). Additionally, the DC link can be modeled with a resistor. Thus, " # Vdci − Vdcj + Idci ∗ Rdc f (x) = . (5.28) Idci + Idcj The control strategy for a point-to-point VSC-HVDC link needs to take both converter substations into account. Since only one DC voltage reference can be used in the link, either rectifier or inverter would be assigned to control the DC voltage, and the other one has to choose the active power control for the d axis control. With respect to the q axis control, both substations can be operated under AC voltage

5.3. VSC-HVDC

71

control or reactive power control. Moreover, from each converter’s point of view, (5.26) still holds for d and q axis controls. Therefore, by combining (5.27) and (5.28) the point-to-point VSC-HVDC link model can be formulated.

5.3.4

Case study

The 6-bus test system is developed in matlab/Simulink with a point-to-point VSC-HVDC link placed between bus 3 and bus 4. Average value models are used for the VSCs. A circuit breaker located on line 4, between bus 4 and 6, was opened at t = 2 s and after three cycles it was re-closed at t = 2.06 s. Figure 5.15 shows partial estimation results for multiple snapshots. 1.05 Vmag−true Vmag−meas. Vmag−est.

0.9

Vrdc(p.u.)

|V|(p.u.)

1

Hvdc−true Hvdcm Hvdc−est

1 0.95

0.8 1

1.5

2

2.5 Time (s)

3

3.5

4

1

0

2

2.5 Time (s)

3

3.5

4

0

10

10 Error (p.u.)

Error (p.u.)

Vmag−estimation−residual

1

Vrdc−residual−error

−5

10

−5

10

1.5

1.5

2

2.5 Time (s)

3

3.5

4

1

Idc (p.u.)

(a) |V˜ | at bus 4

2

2.5 Time (s)

3

3.5

4

(b) DC voltage on the rectifier side

1 Hvdc−true Hvdcm Hvdc−est

0.8 0.6 1

Error (p.u.)

1.5

1.5

2

2.5 Time (s)

3

3.5

4

−2

10

Idc−residual−error

−4

10

1

1.5

2

2.5 Time (s)

3

3.5

4

(c) DC current

Figure 5.15: NSE for the 6-bus hybrid AC/DC system with a point-to-point VSC-HVDC link for multiple snapshots As shown in Fig. 5.15, the residuals between estimation results and “true values” (obtained from matlab/Simulink simulation and then processed to mimic phasor data) are maintained around 10−3 , which is reasonable in the presence of ambigu-

72

Nonlinear network models

ous modelings between matlab/Simulink model and state estimator model. This simulation study mimics a realistic condition that may be found in field installations. On the other hand, when the system was subject to a perturbation, the estimation residuals increased to 10−1 . This is mainly due to the lack of representation of the transient dynamics in the estimation model. Thus even when a system is oscillating after a perturbation occurs, the estimation model remains the same and is not able to take the system’s dynamical behavior into account properly. How to consider the system dynamics when operating a static state estimation becomes an intriguing problem and will be introduced in next chapter.

5.4

FACTS

FACTS has become an indispensable asset to improve transmission quality and efficiency of existing AC grids. Therefore, FACTS devices have to be properly modeled and studied for power system operation and control. Depending on the types of power electronic components being used, FACTS devices can be classified as thyristor-based and converter-based devices. Thyristor devices have no gate turn-off capability while converter devices, i.e. voltage source converters (VSCs) and current source converters (CSCs), have gate turn-off capability. This characteristic results in differences in control schemes. Thyristor-based FACTS include devices such as static var compensator (SVC) and thyristor controlled series compensator (TCSC); converter-based FACTS include devices such as static synchronous compensator (STATCOM) and static synchronous series compensator (SSSC). Both a thyristor-based and a converter-based devices with a similar control function can have a similar response within their linear operating range (under steady state conditions). Therefore, it is reasonable to categorize FACTS devices according to their functions. To this end, FACTS can be categorized as shunt devices (e.g., SVC and STATCOM), series devices (e.g., TCSC and SSSC), and combined series-series or series-shunt devices (e.g., UPFC). This section will briefly present the network models of SVC, STATCOM, TCSC and SSSC for PMU-based state estimations. Other FACTS models can be easily obtained by modifying or extending these models with minor changes.

5.4.1

Shunt devices

A shunt FACTS device serves as a static compensator such as SVC and STATCOM. The compensator varies its reactive output power to control the voltage at given terminals of the transmission network so as to maintain the desired power flow under possible system disturbances. This voltage control is performed on the bus independently from the individual lines connected to it [27]. In addition, as shunt devices are not embedded in transmission lines, they do not need to sustain contingencies and dynamic overloads compared to series devices. On the other hand,

5.4. FACTS

73

shunt devices do not have the advantage of controlling power flow on lines and they are bigger than series devices for a required MVA size. In practice, SVC and STATCOM are not used as a perfect voltage regulator 1 (VT = Vref ), but rather a droop controller (Ish = Slope (VT − Vref )), where the terminal voltage can vary in proportion with the compensating current. This is represented in Fig. 5.16 by the linear slope between the terminal voltage and the compensating current before the current hits its capacitive or inductive limit. For terminal voltages that are out of the linear control range, the compensating current of STATCOM stays at the maximum capacitive or inductive value; in contrast, SVC changes in the manner of a fixed capacitor or inductor. Our static estimation models only take into consideration the linear control range.

VT STATCOM

SVC

Vref

SVC

STATCOM

Capacitive

I sh

Inductive cap I max

ind I max

I sh

Figure 5.16: V-I characteristic of the SVC and the STATCOM [27]

5.4.1.1

Static Var Compensator

SVC is a controlled variable reactive impedance type var generator, which employs thyristor-controlled (TCR) reactors with fixed and/or thyristor-switched capacitors (FC and/or TSC). Figure 5.17a and 5.17b shows basic diagrams of FC-TCR and TSC-TCR, respectively. The firing angle (α) controls the turn-on period of the thyristor so as to vary the equivalent reactance of the SVC. To simplify the network model and reduce the number of state variables, the controlled variable is assumed to be an equivalent susceptance, bSV C [78], as shown in Fig. 5.17c. Its network model can be formulated as: f (x) =

h

|V˜i | − Viref − Kbsvc

i

,

(5.29)

where Viref is the constant voltage which the SVC aims to maintain for bus i and V˜i is the bus voltage phasor; bsvc is the equivalent susceptance value. K is the slope of the V-I linear characteristic, typically 1 − 5%. Therefore, the states of an AC

V

74

Nonlinear network models

Bus i bSVC

V

V

(a) Basic FC-TCR type static (b) Basic TSC-TCR type static (c) equivalent susceptance var generator var generator model

Figure 5.17: SVC model schemes [27, 78] system with SVCs can be denoted as V   ˜ |˜I|, θ, δ, bsvc T , x = |V|,

(5.30)

˜ |˜I|, θ and δ are AC system states. where |V|, 5.4.1.2

Static Synchronous Compensator

STATCOM is a synchronous voltage source type static var generator which employs a switching power converter. It can be VSC-based, as shown in Fig. 5.18a, or CSCbased. Its model can be simplified as a current injection source for state estimation purposes, as shown in Fig. 5.18b. The STATCOM current I˜st is always in quadrature with the bus voltage V˜i , hence only reactive power is drawn or injected into the connected bus. Its network model can be formulated as: h i f (x) = |V˜i | − Viref − K|I˜st | , (5.31) where I˜st is the equivalent current phasor generated by STATCOM; other notations are the same with those used in SVC model. Therefore, the states of an AC system

Bus i

Bus i Ist

(b) equivalent current source model (a) VSC-based STATCOM

Figure 5.18: STATCOM model schemes [27, 78]

5.4. FACTS

75

with STATCOMs can be denoted as   ˜ |˜I|, θ, δ, |˜Ist | T , x = |V|,

(5.32)

˜ |˜I|, θ and δ are AC system states. where |V|, Both SVC and STATCOM operate to maintain the bus voltage by manipulating the reactive power, but in different ways: SVC is modeled as an equivalent susceptance bsvc while STATCOM as a current source I˜st .

5.4.2

Series devices

The shunt compensation is effective in maintaining the desired voltage profile at buses. However, it is ineffective in controlling the actual transmitted power at a defined transmission voltage. In contrast, series capacitive compensation is able to cancel a portion of the reactive line impedance and thereby increases the transmittable power. On the other hand, series FACTS devices have smaller size for the same MVA rating of a shunt device, but they have to be designed to be able to sustain different contingencies and overloads. In general, series FACTS can be classified into thyristor-control impedance type (e.g., TCSC) and converter-based voltage source type compensators (e.g., SSSC). 5.4.2.1

Thyristor Controlled Series Compensator

A TCSC consists of the series compensating capacitor shunted by a thyristorcontrolled reactor, as shown in Fig. 5.19a. It allows varying the series reactance of a transmission line to regulate the active power flow through the line. Although firing angle (α) is the control variable, an equivalent susceptance btc is used in order to simplify the network model and reduce the number of states variables, as shown in Fig. 5.19b. Thus, the network model of TCSC can be formulated as: h i f (x) = Pijref − |V˜i ||V˜j |(yij + btc ) sin(θi − θj ) , (5.33) where Pijref is the constant active power flow reference on which the TCSC aims to maintain; yij is the line admittance without considering the TCSC and btc is the

α Bus i

yij

Bus j

Bus i

bTCSC

yij

Bus j

α (b) equivalent susceptance model (a) firing angle model

Figure 5.19: TCSC model schemes [27, 78]

76

Nonlinear network models

Line ij

Line ij

 I ijI ij VVSSSC SSSC Busi i Bus

Zyij ij

BusBus j j

(b) equivalent synchronous voltage source model (a) VSC-based SSSC

Figure 5.20: SSSC model schemes [27, 78, 108] equivalent susceptance of the TCSC; θi and θj are the angles of voltage phasors V˜i and V˜j , respectively. Similarly to (5.29) and (5.31), (5.33) is added to the network model. In addition, the equations for the AC line where the TCSC is installed also need to be updated accordingly as the line impedance changes. It is notable that a TCSC can be operated under different control modes. Maintaining a constant active power is a commonly used one. Apart from that, maintaining specific impedance level, or current magnitude through the line is also feasible [107]. This network model is developed based on the constant active power control mode. For other control modes, it needs minor modification. 5.4.2.2

Static Synchronous Series Compensator

An SSSC is a converter-based FACTS device as shown in Fig. 5.20a. Its model can be simplified as a series voltage source [27, 78, 108] for state estimation purposes, as shown in Fig. 5.20b. An SSSC is able to provide a constant compensating voltage independently from variable line currents. This voltage V˜ss is always in quadrature with the line current I˜ij , hence only its magnitude is controllable. In some cases, the injected voltage is assigned larger than the voltage difference between the sending- and receiving-end, that is, if |V˜ss | > |V˜i − V˜j |, the power flow will reverse. This bi-directional compensation capability distinguishes SSSCs from other series devices. However, in many practical applications, only capacitive series line compensation is required. An SSSC can be controlled in three different ways: • in voltage compensation mode [27, 78], the SSSC can maintain the rated capacitive or inductive compensating voltage regardless of the changes in the line current. This voltage V˜ss is always in quadrature with the line current I˜ij , hence only its magnitude is controllable. • in impedance compensation mode [27, 78], the SSSC is fixed to the maximum rated capacitive or inductive compensating reactance.

5.4. FACTS

77

• some literature (e.g. [108]) also proposes to use SSSC for controlling the active power flow of the line to a desired value. In all the control methods, the active power injected into the transmission line by the SSSC is zero unless the series VSC has an energy storage system connected to the DC capacitor or it is connected to the DC bus of another VSC. Under constant voltage control mode, an SSSC can be formulated as:   (±V˜ss + V˜i − V˜j ) ∗ yij − I˜ij   ref f (x) =  |V˜ss | − Vss (5.34) , θss − δij − π/2 ref where V˜ss is the equivalent synchronous voltage of an SSSC and Vss is the constant voltage value which the SSSC aims to maintain; θss and δij are the angles of SSSC equivalent voltage V˜ss and line current I˜ij , respectively. Note that the ± symbol indicates the type of compensating voltage: a positive symbol indicates a capacitive compensation and a negative symbol indicates an inductive compensation. For other control modes, this model needs to be adapted. Equation (5.34) replaces the corresponding (row) equations that are associated with the SSSCs in the AC network model, hence, the number of network equations remains the same. As phasors in the proposed state estimator are represented by magnitudes and angles, (5.34) is thus rewritten as follows:

 f (x) =

|V˜ss |yij cos(δij + π/2 + γij ) + |V˜i |yij cos(θi + γij ) − |V˜j |yij cos(θj + γij ) − |I˜ij | cos(δij ) |V˜ss |yij sin(δij + π/2 + γij ) + |V˜i |yij sin(θi + γij ) − |V˜j |yij sin(θj + γij ) − |I˜ij | sin(δij )

(5.35) where θ and δ are the angles of the voltage phasor and the line current phasor, respectively; γ is the angle of the line admittance.

5.4.3

Case study

The 9-bus test system is used for this case. Shunt devices are installed at bus 8, and series devices are installed on the line between bus 5 and bus 4. 5.4.3.1

Test for the SVC model

An SVC was installed at bus 8, where a 16.67% load increase (both active power and reactive power) was applied at t = 2 s. Figure 5.21 shows the estimation results for multiple snapshots. As shown in Fig. 5.21, the estimation residuals for both the voltage magnitude at bus 8 and the equivalent susceptance reach below 10−13 during steady state. When the perturbation occurs, the estimation residuals of the voltage magnitude manage to maintain on the same level as before, which is due to the availability of its measurement. However, the estimation residuals of the equivalent susceptance

 ,

78

Nonlinear network models

Vmag−true Vmag−m Vmag−est

1.025 1.02

1

1.5

2

2.5

3

3.5

0.4

bsvc(p.u.)

|V|(p.u.).

1.03

x 10

2

2.5

3

3.5

4

0

10 Vmag−residual−error

0.5

error(p.u.)

error(p.u.)

1.5

time

−14

0

0.2 0.1 1

4

time 1

bsvc−true bsvc−m bsvc−est

0.3

−10

10

bsvc−residual−error −20

1

1.5

2

2.5

3

3.5

10

4

1

time

1.5

2

2.5

3

3.5

4

time

(a) |V˜ | at bus 8

(b) bsvc

Figure 5.21: NSE for the 9-bus system with an SVC for multiple snapshots Vmag−true Vmag−m Vmag−est

1.025 1.02 1

1.5

2

2.5

3

3.5

0.4

|Ist|(p.u.)

|V|(p.u.).

1.03

|Ist|−ture |Ist|−est 0.2

0

4

1

1.5

2

time −15

3.5

4

10 Vmag−residual−error

4 2 0

3

0

x 10

Error(p.u.)

Error(p.u.)

6

2.5

time

−10

10

|Ist|−residual−error −20

1

1.5

2

2.5

3

3.5

4

10

time

(a) |V˜ | at bus 8

1

1.5

2

2.5

3

3.5

4

time

(b) Ist

Figure 5.22: NSE for the 9-bus system with a STATCOM for multiple snapshots are subject to a big leap and then come down to about 10−5 . This big leap of the estimation residual implies the limited performance of the static state estimation when the system is under large dynamic changes and power electronic devices react to these changes. 5.4.3.2

Test for the STATCOM model

The same scenario used for the SVC test was also applied to the test system with a STATCOM installed at bus 8. As shown in Fig. 5.22, the estimation performs relatively close with that in the SVC test.

5.4. FACTS

79

|ytcsc|(p.u.)

|V|(p.u.).

1.05 1

Vmag−true Vmag−m Vmag−est

0.95 0.9 0.85 1

1.5

2

2.5

3

3.5

40 20 1

4

ytcsc−true ytcsc−m ytcsc−est

60

1.5

2

3

3.5

4

0

100

10

Vmag−residual−error

− 10 10

− 10 20 1

|ytcsc|(p.u.)

Error (p.u.)

2.5 time

time

−10

ytcsc−residual−error

10

−20

1.5

2

2.5

3

3.5

10

4

1

1.5

2

2.5

3

3.5

4

time

time

(a) |V˜ | at bus 5

(b) ytc

Figure 5.23: NSE for the 9-bus system with a TCSC for multiple snapshots 0.6

1

Vmag−true Vmag−m Vmag−est

0.95 0.9 0.85 1

1.5

2

2.5

3

3.5

|I|(p.u.)

|V|(p.u.).

1.05

0.4 1

4

Imag−true Imag−m Imag−est

0.5

1.5

2

3

3.5

4

0

100

10

Vmag−residual−error

− 10 10

− 10 20 1

|I|(p.u.)

Error (p.u.)

2.5 time

time

Imag−residual−error

−10

10

−20

1.5

2

2.5

3

3.5

4

time

(a) |V˜ | at bus 5

10

1

1.5

2

2.5

3

3.5

4

time

˜ on the line between bus 5 and bus 4 (b) |I|

Figure 5.24: NSE for the 9-bus system with an SSSC for multiple snapshots 5.4.3.3

Test for the TCSC model

A TCSC was installed on the line between bus 5 and bus 4, and a fault was applied at bus 3 from t = 2 s to t = 2.04 s. Figure 5.23 shows the estimation results for multiple snapshots, where ytc denotes the line admittance considering the TCSC. The residuals increase immediately at the instance when the fault was applied, from around 10−13 to 10−3 .

5.4.3.4

Test for the SSSC model

The same scenario for TCSC was applied on the test system with an SSSC installed on the line between bus 5 and bus 4. Figure 5.24 shows the estimation results for multiple snapshots. The residuals increase immediately at the instance when the fault was applied, from around 10−13 to 10−3 .

80

5.5

Nonlinear network models

Summary

This chapter presents a novel way of formulating the measurement model of a PMUbased state estimator, where the network model is separated from measurements in order to protect the network model from missing measurements and assign different weights to them separately. Sequentially, network models of classic HVDC link, VSC-HVDC link, and FACTS are introduced. The proposed network model simplifies the nonlinearities of the conventional network model. Different control modes are also included to enrich network information. All the AC and DC states are simultaneously considered to solve nonlinear WLS problem. After presenting the network models for the state estimator, case studies for classic HVDC link, VSC-HVDC link, and FACTS are conducted individually to validate the proposed network models. At the same time, it is noted that the estimation accuracy needs to be improved when the system is under large dynamic changes. Testing results indicate that additional modeling details may need to be included to obtain higher accuracy when the system is under large dynamic changes and power electronic devices react to these changes. This will be addressed in the following chapter.

Chapter 6

Pseudo-dynamic network modeling and examples As renewable energy integration brings intermittent fluctuations into power systems, state estimation applications face a greater need to capture system dynamics in a faster and more flexible way, which has been difficult for the static state estimations. On the other hand, most of the dynamic state estimations and forecasting-aided state estimations are computationally demanding, which introduces delays in the estimation calculation cycle [109]. When the calculation cycle is longer than the PMU data rate, the estimated results become less valuable for on-line applications, especially those with real-time requirements. Therefore, we propose a pseudo-dynamic modeling approach [110] that can improve the estimation accuracy during transients without significantly increasing the estimation’s computational burden. This pseudo-dynamic modeling approach offers the following advantages: • High accuracy: it establishes difference equations from the dynamic model of certain components from both AC and DC grids, and then combines them with the static network equations. This process is named as pseudo-dynamic network modeling. • Compatible modeling and implementation: the pseudo-dynamic network model maintains the basic structure of the static model, which greatly reduces the workload of re-composing network models when updating existing algorithms. • Fast computation: the algorithm used to solve static state estimations (i.e.,WLS) is directly applied to the pseudo-dynamic state estimation, which ensures that the computational speed will be minimally affected. • Explicit representation of control modes: for devices with time-varying control references, the pseudo-dynamic state estimation is capable of taking their control modes into account, which would be challenging for static-only state estimations. 81

82

Pseudo-dynamic network modeling and examples

At the same time, as the power electronics-based devices (e.g., FACTS and VSC-based HVDC) play an important role in improving the system control ability and flexibility with the increased integration of renewable energy, their real-time performance during transients needs to be monitored in order to make full use of these devices in on-line operations. To this end, suitable pseudo-dynamic models that can represent power electronics-based devices for both steady state and transient conditions are developed. Two power electronics-based devices are used to illustrate the proposed approach—STATCOM, as an example of a FACTS device, and VSCHVDC link.

6.1

Pseudo-dynamic concept

Network models for static state estimations describe the system topology and properties under normal operating conditions, which can be considered to be in the quasi-stationary regime. However, when the system is under transient conditions, the static network model for certain components may no longer hold. If the models for such components are not replaced by the models that can represent their dynamic behaviors, the estimation error would be large. In other words, when the system enters a transient condition due to a perturbation, if the system cannot restore a steady state before the next PMU snapshot comes, the static network model will conflict with the PMU measurement, leading to an inaccurate state estimation solution. Therefore, a new type of network model, called herein the pseudo-dynamic network model, is proposed. It leverages the existing body of the network model and includes the difference equations that describe the system dynamical properties. The following section explains how to formulate the pseudo-dynamic network model. Typically, it is assumed that power system as a continuous dynamical system can be described by employing ordinary differential equations (ODEs). Even for a higher-order system, its higher-order ODEs can be converted into a larger set of first-order equations by introducing extra variables. However, for PMU-based state estimations, PMU data is streamed discretely over fixed and synchronous time intervals. Therefore, a power system can be treated as a discrete dynamical system, where difference equations are used to update the state variables in discrete time steps of the same size as the PMU data. This discretization is similar to numerically solving differential equations, i.e. numerical integration using Euler’s method, where the states are updated when knowing the starting point and the slope at it, and the error can be made small if the step size is small enough and the interval of computation is finite. However, Euler’s method is insufficiently robust, and thus the Euler’s full-step modification, which belongs to a second-order Runge-Kutta method, is used next. The difference equation used herein is formulated as bk ≈ xk−1 + x

Ts b˙ k ), (x˙ k−1 + x 2

(6.1)

6.2. STATCOM model

83

where Ts is the sample time (step size). x˙ = g(x) can be either a linear or a nonlinear function of x, which is essentially the differential equation of the continuous dynamical system. The value of x˙ k−1 is calculated by substituting xk−1 into g(x), which is denoted by g(x)k−1 . Equation (6.1) implies that the current value is determined by adding the previous value to the average increment during the time interval using the average of the slopes. In order to comply (6.1) with the generalized form of the network model equation f (x), it is rewritten as bk − xk−1 − f (xk ) : x

 Ts  g(x)k−1 + g(x)k . 2

(6.2)

The foregoing is the procedure of pseudo-dynamic network modeling and (6.2) is a pseudo-dynamic network model, which can be used to describe all different components with dynamical behavior. While the majority of the literature considers the use of Kalman Filters or other types of observers for a similar purpose, this work chooses a simpler approach, which only requires few PMU measurements close to a certain component and without the need of formulating a complex dynamic model. Two examples are presented in the following sections.

6.2

STATCOM model

This section introduces the pseudo-dynamic network model of a STATCOM. It is used as an example of showing how to develop a pseudo-dynamic network model for a controller.

V ref V

 

K T  s 1

I st

Figure 6.1: Simplified STATCOM control block diagram A simplified STATCOM’s control process can be represented by the block diagram shown in Fig. 6.1. The output is the current magnitude |Ist |, which varies the current flow at the connected bus in order to change the reactive power flow and control the bus voltage. Ist is perpendicular to the bus voltage phasor, hence its angle can be computed from the voltage angle. This controller model is given by |I˙st | =

K ref 1 (V − |V |) − |Ist |. T T

(6.3)

Using the pseudo-dynamic network model described by (6.2), (6.3) can be rewritten as: f (xk ) : (1+

Ts K b Ts K ref Ts Ts K Ts b )|Ist,k |+ |Vk |− V −(1− )|Ist,k−1 |+ |Vk−1 |. (6.4) 2T 2T T 2T 2T

84

Pseudo-dynamic network modeling and examples

Equation (6.4) embodies the dynamic relation between |Vb | and |Ibst |, and replaces the static relation in (5.31). This pseudo-dynamic network model of STATCOM will be validated using the real PMU data in Sec. 6.4.1, and simulation studies in Sec. 6.4.2. The states of the new model are the same as those of the static model (5.32). The corresponding Jacobian matrix elements must be updated using

H(xk ) :

6.3 6.3.1

∂h(xk ) Ts ∂h(xk ) Ts K , . =1+ = b b 2T 2T ∂|Ist,k | ∂|Vk |

(6.5)

VSC-HVDC model VSC substation model

In (5.24), the modulation index Mv defines the relation between Vdc and |Vv |. However, in reality |Vv | varies with the system dynamics; while Mv is a fixed value that depends on the control mode. Therefore, in order to improve the accuracy of the model especially during transients, |Vv | in (5.24) is replaced by |v ref |, which is the voltage reference of the converter’s bridges. This voltage reference is generated by the converter’s control system. The substation’s pseudo-dynamic model intends to represent the control process of the converter. As already explained in Section 5.3.2, the most common high-level control strategy for VSCs is vector-current control, which has a two-level control scheme, so-called the outer active-reactive power and voltage loop (here abbreviated to outer loop), and the inner current loop (here abbreviated to inner loop). As shown in Fig. 6.2, the outer loop transfers the VSC control references, i.e., P ref , ref ref ref T ref Qref , Vdc and Vac , into the converter’s current references, iref iq ) . In dq = (id this loop, three-phase fundamental currents and voltages are transformed into dq components in a synchronously rotating reference frame through Clark’s and Park’s transformations. Hence, all quantities become DC signals [111]. Depending on the converter’s operation mode, reference iref can be determined d ref ref by either active power P or DC voltage Vdc . Similarly, reference iref can be q ref determined by either reactive power Qref or AC voltage Vac at the PCC. For each VSC substation, only one iref and only one iref can be utilized. An ordinary q d integral controller can be used for outer loop, which can be formulated using the

6.3. VSC-HVDC model

85

pseudo-dynamic equations (6.2) as:

f (xk )

:biref d,k

or biref d,k biref q,k or biref q,k

! P ref − Pbk−1 P ref − Pbk + − , |Vf,k−1 | |Vbf,k |  Ts  ref b 2V − V − V ; − iref − K dc,k dc,k−1 V dc dc d,k−1 2 ! b k−1 bk Ts Qref − Q Qref − Q ref − iq,k−1 − KQ + , 2 |Vf,k−1 | |Vbf,k |  Ts  ref b 2Vac − Vac,k − Vac,k−1 ; − iref q,k−1 − KV ac 2 iref d,k−1

Ts − KP 2

(6.6)

b k = |Vbk ||Ibk | sin(θbk − δbk ). where Pbk = |Vbk ||Ibk | cos(θbk − δbk ), Q Next, the outputs of the outer loop become the inner loop’s input. The inner ref ref T loop transfers the current references iref iq ) into the voltage references dq = (id ref ref T ref of the converter’s bridges vdq = (vd vq ) , which then are transformed into the three-phase voltage references.

Pref

I dref

ref dc ref

V Q

Vdref

Varef

Vbref

I qref

ref ac

V

Outer loop control

Vqref

Vcref

Inner loop control Transformation

Figure 6.2: The vector-current control process Based on the converter’s structure shown in Fig. 5.13, the fundamental voltage of the converter’s bridges equals to the voltage at PCC minus the voltage drop on the transformer and the phase reactor. By neglecting the resistances of the transformer and the phase reactor, an ordinary PI controller can be used for the inner loop, which is formulated as: 1 v ref = V − (Kp + Ki )(iref − Iv ) − j(Xt + Xp )iref , s

(6.7)

where v ref = vdref + jvqref and iref = iref + jiref q . Equation (6.7) can be rewritten d as:    1  ref vkref = Vk − Kp iref − I − K i − I − j(Xt + Xp )iref v,k i v,k k k . s k

(6.8)

86

Pseudo-dynamic network modeling and examples

In order to formulate (6.8) in the form of the pseudo-dynamic network model, 1 assume that ybk = Ki (biref − Ibv,k ), then s k  Ts bref b ybk = yk−1 + Ki ik − Iv,k + iref − I , v,k−1 k−1 2 ref ref yk−1 = −vk−1 + Vk−1 − Kp (iref k−1 − Iv,k−1 ) − j(Xt + Xp )ik−1 . Substituting ybk and yk−1 into (6.8), then   ref bv,k − iref + Iv,k−1 f (xk ) =b vkref − vk−1 − Vbk + Vk−1 + Kp biref − I k k−1     Ts bref b ref ref b ik − Iv,k + iref − I + j(X + X ) i − i + Ki v,k−1 t p k−1 k k−1 . 2 (6.9) As all the states at step (k − 1) are known, they can be considered as a constant component that is denoted by C. Hence, (6.9) can be simplified as   Ts bref b  f (xk ) = vbkref − Vbk + Kp + Ki ik − Iv,k + j(Xt + Xp )biref + C. (6.10) k 2 The above equation can be split into d and q components: 

f (xk ) = 

 
 ref b b bi bref d,k − |Iv,k | cos(δv,k ) − (Xt + Xp )iq,k + Cd  .
 b b bref − |Vbk | sin(θbk ) + Kp + Ki T2s biref q,k − |Iv,k | sin(δv,k ) + (Xt + Xp )id,k + Cq

ref vbd,k − |Vbk | cos(θbk ) + Kp + Ki T2s ref vbq,k

(6.11) Thus, the voltage reference of the converter’s bridges vref can be expressed by (6.6) and (6.11). Consequently (5.24) becomes q " # Kd2a Mv Vdc − (vdref )2 + (vqref )2 f (x) = . (6.12) (Kd2a )2 Vdc Idc − |Vv ||Iv | cos(θv − δv ) Therefore, (5.21), (6.12), (6.6), and (6.11) compose the pseudo-dynamic network model for VSC substation. As a result, four more states are added for the pseudodynamic model in addition to the states of the static model 5.25: ref ref ref T x = [|V|, |I|, θ, δ, |Vf |, θf , |If |, δf , |Vv |, θv , Vdc , Idc , iref d , iq , vd , vq ] .

(6.13)

The Jacobian matrix needs to be updated w.r.t. (6.6), (6.11), and (6.12). Taking the first equation of (6.6) as an example, its corresponding Jacobian matrix element w.r.t. voltage magnitude is given by H(xk ) :

∂f (xk ) Ts |Ibk |cos(θbk − δbk ) = KP . 2 ∂|Vbk | |Vbf,k |

6.4. Case study

87



I cc

Cdc1 V dc1

Rdc

Cdc1

Rdc

Ldc



I dc1



Vdc 2 Cdc 2 

Ldc

I dc 2

Cdc 2

I cc Figure 6.3: DC circuit of a point-to-point VSC-HVDC link model

6.3.2

Point-to-point VSC-HVDC link model

Using the VSC substation model constructed above, other VSC models can be developed, among which point-to-point VSC-HVDC link is of great interest. To model it, one VSC substation acts as the rectifier and the other one as the inverter, depending on which side controls the active power flow. At each substation a large DC capacitor is installed, and a DC cable connects two substations [112]. The DC circuit of a point-to-point VSC-HVDC link is shown in Fig. 6.3 and the basic equations are:    f (x) =   

dVdc1 − (Idc1 − Icc ) dt dVdc2 Cdc2 − (Idc2 + Icc ) dt dIcc Ldc − (Vdc1 − Vdc2 − Rdc Icc ) dt

Cdc1

   .  

(6.14)

The corresponding pseudo-dynamic equations are: 

s s s s Vbdc1,k − 2CTdc1 Ibdc1,k + 2CTdc1 Ibcc,k − Vdc1,k−1 − 2CTdc1 Idc1,k−1 + 2CTdc1 Icc,k−1  b Ts b Ts b Ts Ts f (xk ) =  Vdc2,k − 2Cdc2 Idc2,k − 2Cdc2 Icc,k − Vdc2,k−1 − 2Cdc2 Idc2,k−1 − 2Cdc2 Icc,k−1 Ts b Ts b Ts s Rdc b s Rdc (1 + T2L )Icc,k − 2L Vdc1,k + 2L Vdc2,k − (1 − T2L )Icc,k−1 − 2L Vdc1,k−1 + dc dc dc dc dc

 Ts 2Ldc Vdc2,k−1

(6.15)

6.4

Case study

This section focuses on studying the performances of the PMU-based state estimation when using the static network model and the proposed pseudo-dynamic network model. Subsection 6.4.1 uses real PMU data to compare and validate these two models in the case of a STATCOM. In Subsection 6.4.2, two test systems with a STATCOM installed are built up and simulated in PSAT to generate the synthetic measurements for the accuracy comparison. At last, the VSC-Based HVDC Link model provided by matlab/Simulink is used to generate the synthetic measurements for two test scenarios in Subsection 6.4.3.

 .

88

6.4.1

Pseudo-dynamic network modeling and examples

STATCOM models’ comparison and validation using real PMU data

The real PMU data used in this subsection was recorded during a generator trip event. The generator loss resulted in reducing the active power flow on the main transfer paths in a neighboring system, and caused an increase in bus voltages. A STATCOM installed on the transfer paths reacted to the voltage change. The STATCOM bus voltage phasor, as well as its output current phasor, was measured by a PMU with a reporting rate of 30 samples/second. Figure 6.4 shows the voltage magnitude, current magnitude and angle differences. STATCOM voltage magnitude Vm [pu]

1.04

1.02

1 0

50

100

150

200

250

300

time [s] STATCOM current magnitude Im [pu]

0.5

0

−0.5 0

50

100

150

200

250

300

time [s] STATCOM VI angle differences θ [deg]

100

0

−100 0

50

100

150

200

250

300

time [s]

Figure 6.4: The STATCOM’s PMU measurements: voltage magnitude (top); current magnitude (middle); angle difference between voltage and current (bottom) The PMU data is used to compute the STATCOM’s V-I curve as shown in Fig. 6.5. Blue dots represent pre-fault operation points, which depict a linear V-I characteristic. When the fault occurred, several green dots scatter away from the V-I characteristic. Gradually the system reached another stable operation point where the STATCOM current is about 0.25 p.u. Note that the STATCOM’s output

6.4. Case study

89 1.04

Vm [pu]

1.035

1.03

1.025

Pre−fault transient oscillation reference changes new operation point var reserve V−I characteristics

1.02

−0.2 −0.1

0

0.1

0.2

0.3

0.4

0.5

Im [pu]

Figure 6.5: The STATCOM’s V-I curve computed using the PMU data current switched from capacitive to inductive after the disturbance occurred, aiming to decrease the bus voltage. When the system was approaching the next steady state, the system operator changed the voltage reference of the STATCOM in several steps, which can be seen from the saw-tooth variation in red (the voltage reference corresponds to the voltage when the STATCOM current output equals to zero). The system reached a new operation point after the reference changed, which is represented by the light blue dots. As the system turned to a new steady state condition, the system operator switched the STATCOM to the var reserve control mode such that other slower voltage controls in the system can take over the reactive power support. Figure 6.5 shows the variation of operating conditions during the whole event and clearly reveals the linear V-I droop relation of a STATCOM in steady-state operation. Using the pre-fault PMU data (0 to 69.33s), this linear V-I characteristic was determined by applying the linear regression function polyfit in matlab, which is shown by the black line in Fig. 6.5. The two coefficients of the linear predictor, slope and the intercept, represent the equivalent impedance and voltage reference of the STATCOM, respectively. For the pre-fault steady state, Xs = 0.0285 and V ref = 1.0214. Note that during the transients, the operating points scatter and do not exactly follow the V-I characteristic. For a certain |V |, the measured |Ist | and the read value from the V-I characteristic can deviate up to 0.04 p.u. This implies that the static STATCOM model, which is based on the V-I characteristic, would result in deviations during transients. On the other hand, the pseudo-dynamic model can be a good choice to reflect the STATCOM’s control process and its system dynamics. Using the transfer function 32.1981 estimator tfest in matlab, the transfer function is estimated as 0.0329s+1 , with

90

Pseudo-dynamic network modeling and examples

two control parameters K = 32.1981 and T = 0.0329, which are those in (6.3)-(6.5). The gain K theoretically is the reciprocal of the V-I characteristics’ slope, which has been calculated for the pre-fault steady state, i.e. Xs = 0.0285. Its inverse (32.1543) is quite close to the calculated K (32.1981), which verifies that of the transfer function. In addition, the time constant T is in accordance with the statement of STATCOM in [27], which says “typically about 10-50 ms depending on the var generator transport lag”. In order to intuitively compare the STATCOM’s static model with its pseudodynamic model, PMU data of the voltage magnitude |V | is used as the arbitrary inputs of the static model (see (5.31)) and the pseudo-dynamic model (see (6.3)-(6.5)), respectively. The static model applies the STATCOM’s linear V-I characteristic and the pseudo-dynamic model applies its first-order control model whose parameters were obtained above through model identification. These models’ outputs are then compared to the |Ist | PMU measurement in Fig. 6.6. 0.6 PMU measurement pseudo−dynamic model output static model output

0.5 |Ist| [pu]

0.4 0.3 0.2 0.1 0 −0.1 69.5 70

70.5 71

71.5 72 72.5 73 time [s]

73.5 74

Figure 6.6: PMU data, static model output and pseudo-dynamic model output for the current at the STATCOM Figure 6.6 compares results: the pseudo-dynamic model’ output |Ist | coincides with the PMU data; while the static model’s output gradually deviates from the the PMU data. The reason that the deviation is not prominent is because the time constant of the STATCOM is quite close to the PMU sampling rate, thereby its dynamical trajecory can be nearly tracked by the PMU measurement. For a larger and more complex system, this deviation could be amplified further. An analysis of their residuals w.r.t. the PMU data is shown in Table 6.1.

6.4.2

STATCOM model in two test systems

Next, in order to further study the proposed STATCOM model and PMU-based state estimation algorithm, the 9-bus test system and the KTH-Nordic 32 test system are used. In the modified 9-bus test system, a STATCOM is installed at bus

6.4. Case study

91 Table 6.1: STATCOM models accuracy comparison µ*

δ **

Max. residual

Pseudo-dynamic

0.0012

0.0056

0.0152

Static

0.0198

0.0085

0.0406

SE methods

*

µ denotes the average value of the residuals w.r.t. the PMU data

**

δ denotes the standard deviation of the residuals w.r.t. the PMU data

|Ist|−m |Ist|−est

0.3 0.2 0.1 1.5

2

|Ist|(p.u.)

|Ist|(p.u.)

0.4

|Ist|−m |Ist|−est 0.2

0.1 1.5

2.5

time (s) 10 Error(p.u.)

Error(p.u.)

2.5

−10

0

10

−10

10

|Ist|−residual−error

−20

10

2

time (s)

1.5

2

time (s)

(a) |Ist | Static

2.5

−15

10

|Ist|−residual−error

−20

10

1.5

2

2.5

time (s)

(b) |Ist | Pseudo-dynamic

Figure 6.7: Static and Pseudo-dynamic SEs for the modified 9-bus system 8; in the modified KTH-Nordic 32 test system, the same STATCOM is installed at bus 43. The two key parameters are K = 25, T = 0.04. 6.4.2.1

Pseudo-dynamic SE for the modified 9-bus system

A 16.67% load increase (both active power and reactive power) at bus 8 was applied at t = 2s. As shown in Fig. 6.7, for both cases the current magnitude residuals before the perturbation occurred are below 10−13 p.u.; however, after the instance when the perturbation occurred, the static estimation residuals increase up to 0.1783 p.u. then drop to 10−3 p.u. while the pseudo-dynamic estimation successfully stays on the same accuracy level during transient dynamics. More estimation accuracy performances are shown in Table 6.2. 6.4.2.2

Pseudo-dynamic SE for the modified KTH-Nordic 32 system

A 33% load increase (both active power and reactive power) at Bus 43 was applied at t = 2s. As shown in Fig. 6.8, the static estimation and pseudo-dynamic estimation hold similar estimation accuracy of the current magnitude before the perturbation. However, after the perturbation occurs, the current magnitude estimated by the

Pseudo-dynamic network modeling and examples

0.6 0.4 0.2

|Ist|−m |Ist|−est

0 −0.2 1.5

2

|Ist|(p.u.)

|Ist|(p.u.)

92

0.2

|Ist|−m |Ist|−est

0 2

2.5

Time (s)

0.4

−10

10 Error(p.u.)

Error(p.u.)

0.4

−0.2 1.5

2.5

Time (s)

0.2 |Ist|−residual−error 0 1.5

0.6

|Ist|−residual−error −15

10

−20

2

Time (s)

(a) |Ist | Static

2.5

10

1.5

2

2.5

Time (s)

(b) |Ist | Pseudo-dynamic

Figure 6.8: Static and Pseudo-dynamic SEs for the modified KTH-Nordic 32 system static estimation shows a residual up to 0.3666 p.u. In contrast, the pseudo-dynamic state estimation gives a maximum residual of 5.0626×10−14 p.u. Table 6.2 summaries the results of these tests.

6.4.3

VSC-HVDC model

The VSC-Based HVDC Transmission Link model provided by matlab R2013b/ SimPowerSystems is used herein to generate the synthetic measurements used to validate the proposed VSC-HVDC model and the PMU-based state estimation algorithm. Its rectifier uses active power and reactive power control, and its inverter uses DC voltage and reactive power control. In the case study, all the control parameters preset in the SimPowerSystems model are kept the same in the pseudo-dynamic state estimation model. 6.4.3.1

First test scenario

The inverter’s DC voltage reference dropped from 1 p.u. to 0.95 p.u. at t = 2.1s. As shown in Fig. 6.9, the pseudo-dynamic state estimation performs more accurately by a factor of 1/1000 than the static state estimation not only during transients, but also during steady state. For instance, the voltage magnitudes on the rectifier side estimated by the static state estimation shows a maximum residual up to 0.0279 p.u. while the pseudo-dynamic state estimation gives a maximum residual of 7.6517 × 10−8 p.u. 6.4.3.2

Second test scenario

A three-phase line breaker on the inverter side was opened from t = 2.1s for 0.12s. For this larger perturbation, as shown in Fig. 6.10, the pseudo-dynamic state estimation performs more accurately by a factor of 1/1000 than the static state estimation.

VSC-HVDC test2

VSC-HVDC test1

STATCOM test2

STATCOM test1

(0.0023, 0.0024) (-0.0091, 0.0035) (-0.0012, 0.0010) (0.0108, 0.0041) (0.0023, 0.0024) (-0.0091, 0.0035) (-0.0012, 0.0010) (0.0108, 0.0041)

Res.Vdc

Res.Idc

Res. |Vv |

Res. |Iv |

Res.Vdc

Res. Idc

(-2.06e-16, 6.37e-15)

Res.|Ist |

Res.|Iv |

(-8.71e-17, 7.13e-17)

Res.|V43 |

Res.|Vv |

(1.57e-16, 1.02e-16)

(-3.84e-15, 9.58e-15)

Res.|V8 |

Res.|Ist |

static

(-3.46e-11, 2.00e-09)

(-6.60e-10, 1.09e-07)

(-1.59e-08, 4.37e-07)

(5.45e-10, 4.74e-09)

(-3.46e-11, 2.00e-09)

(-6.60e-10, 1.09e-07)

(-1.59e-08, 4.37e-07)

(5.45e-10, 4.74e-09)

(2.11e-15, 1.83e-15)

(-8.71e-17, 7.13e-17)

(-3.72e-15, 2.01e-15)

(1.57e-16, 1.02e-16)

pseudo-dynamic

Before perturbation (µ, σ)

(0.0068, 0.0307)

(0.0026, 0.0119)

(-0.0059, 0.0174)

(-0.0131, 0.0423)

(0.0148, 0.0045)

(-0.0094, 0.0021)

(-0.0135, 0.0042)

(0.0187, 0.0039)

(0.0048, 0.0332)

(-2.94e-17, 2.12e-16)

(8.03e-04, 0.0127)

(3.85e-16, 1.27e-15)

static

(1.01e-08, 5.23e-08)

(9.57e-07, 9.75e-07)

(1.18e-06, 1.84e-05)

(-1.62e-08, 3.77e-08)

(2.00e-08, 7.62e-09)

(1.05e-08, 1.25e-07)

(5.60e-08, 2.36e-06)

(-5.39e-08, 1.78e-08)

(1.57e-15, 7.16e-15)

(-2.39e-17, 1.95e-16)

(-3.57e-15, 2.16e-14)

(7.25e-16, 1.76e-15)

pseudo-dynamic

After perturbation (µ, σ)

Table 6.2: SE accuracy performance on test systems

0.1418

0.0400

0.0824

0.0207

0.0237

0.0141

0.0214

0.0279

0.3666

2.44e-15

0.1783

2.95e-07

3.44e-06

8.15e-05

1.87e-07

2.81e-08

4.44e-07

1.04e-05

7.65e-08

5.06e-14

2.22e-15

1.05e-13

8.22e-15

pseudo-dynamic

Max. residual

7.11e-15

static

6.4. Case study 93

94

Pseudo-dynamic network modeling and examples 1.15

1.05

1 1

|Vv|(pu)

|Vv|(pu)

1.1

Vvmag−true Vvmag−est.

1.5

2 Time (s)

2.5

1.05 1 1

3

Error(pu)

0.02 0.01

Vdc(p.u.)

1.5

2 Time (s)

2.5

2.5

3

Vvmag−estimation−residual

0 −5 1.5

2 Time (s)

2.5

3

(b) |Vv | Pseudo-dynamic

(a) |Vv | Static 2.6 Vdc−true Vdc−est.

2.5 2.4 2.3 1

2 Time (s)

x 10

−10 1

3

Vdc(p.u.)

Error(pu)

5

Vvmag−estimation−residual

2.6

1.5 −8

0.03

0 1

Vvmag−true Vvmag−est.

1.1

1.5

2 Time (s)

2.5

3

Vdc−true Vdc−est.

2.5 2.4 2.3 1

1.5

2 Time (s)

2.5

3

−7

5 Vdc−estimation−residual

Vdc(p.u.)

Error(p.u.)

0.015 0.01 0.005 0 1

1.5

2 Time (s)

(c) Vdc Static

2.5

3

x 10

Vdc−estimation−residual

0 −5 1

1.5

2 Time (s)

2.5

3

(d) Vdc Pseudo-dynamic

Figure 6.9: Static and Pseudo-dynamic SEs for the VSC-HVDC link model: first test scenario For instance, the DC current estimated by the static state estimation shows a maximum residual up to 0.1418 p.u. while the pseudo-dynamic state estimation gives a maximum residual of 2.9504 × 10−7 p.u. Table 6.2 summaries the results of these tests. As it might be noted that in Figs. 6.7 and 6.8 both static and pseudo dynamic models maintain almost the same level of accuracy during normal operation (order of 1.0 × 10−14 ) while in Figs. 6.9 and 6.10 improvement during normal operation is seen for the pseudo dynamic model. The reason is the pseudo-dynamic model is capable to include internal states that are calculated in real-time, such as the voltage reference. In contrast, static models do not possess such flexibility.

6.5. Computation performance

95 1.5

Ivmag−true Ivmag−est.

1

|Iv|(p.u.)

|Iv|(p.u.)

1.5

0.5 0 1

1.5

2 Time (s)

2.5

Ivmag−true Ivmag−est.

1 0.5 0 1

3

1.5

2 Time (s)

2.5

3

−4

2

Ivmag−estimation−residual

0 1

Idc(p.u.)

2

|Iv|(p.u.)

1.5

2 Time (s)

2.5

1 0 −1 1

3

1.5

0 2 Time (s)

2.5

3

1.5

Idc−true Idc−est.

1.5

2 Time (s)

(b) |Iv | Pseudo-dynamic

(a) |Iv | Static

1

−1 1

x 10

Ivmag−estimation−residual

0.05

Idc(p.u.)

Error(p.u.)

0.1

2.5

Idc−true Idc−est.

1 0.5 0 1

3

1.5

2 Time (s)

2.5

3

−7

4

x 10

Idc−estimation−residual

Idc−estimation−residual

Idc(p.u.)

Error(p.u.)

0.2

0.1

0 1

1.5

2 Time (s)

2.5

3

(c) Idc Static

2 0 −2 1

1.5

2 Time (s)

2.5

3

(d) Idc Pseudo-dynamic

Figure 6.10: Static and Pseudo-dynamic SEs for the VSC-HVDC link model: second test scenario

6.5

Computation performance

One advantage of the proposed pseudo-dynamic PMU-based state estimation is that it does not significantly increase the computation complexity and burden when compared to the static state estimation. By comparing the number of iterations and computation time of the static and the pseudo-dynamic PMU-based state estimations, as shown in Table 6.3, it can be seen that the pseudo-dynamic state estimation performs similarly to the static state estimation in terms of computational speed. These computations were carried out on the VSC-HVDC test scenario I using an ordinary PC with an Intel(R) Core(TM) i7-2640M CPU @2.80GHz and a 8.00 GB RAM, and using matlab R2013b.

96

Pseudo-dynamic network modeling and examples Table 6.3: Computation performance comparison PMU-based SE method

Aver. comp. time per snapshot

Aver. no. of iter. per snapshot

Largest no. of iteration

P-dynamic

4.754 ms

5.465

10

Static

3.115 ms

5.525

11

6.6

Summary

A PMU-based state estimator using a pseudo-dynamic network model is presented herein. This method significantly improves the state estimation accuracy during transients as compared to the static PMU-based state estimation. In contrast to most dynamic state estimation algorithms, it implements an iterative algorithm to update the estimated states instead of solving DAEs, which significantly saves computational resources. Additionally, the pseudo-dynamic network model can be easily constructed from the original static model and represent devices with time-varying control references. In addition, the pseudo-dynamic network models for STATCOMs and VSCHVDCs are developed. These two models also provide valuable and practical insight on how to develop pseudo-dynamic network models for components and controllers in power systems. The case studies provide sufficient evidence that the pseudo-dynamic state estimation algorithm is capable of performing much more accurate estimation during transient dynamics without significantly increasing computational resources as compared to the static state estimations. The STATCOM using real PMU data shows that the proposed modeling approach and the PMU-based state estimation algorithm is applicable when real PMU data from actual power systems is available.

Chapter 7

Quantifying PMU measurement weights The performance of power system state estimation (PSSE) relies on the properties of the model and those of the collected measurements, such as sampling rate, measurement accuracy and variance, etc. Weighting is a practice of accounting for the confidence in the model and in a measurement. Particularly, weighted least squares (WLS) chooses the weights as inversely proportional to the measurement error variances. This selection is straightforward only when the measurement noise is properly quantified in a systematic fashion, which has not been subject to sufficient attention in literature. As mentioned in Chapter 3, WLS regression resembles the maximum likelihood estimators (MLE) [16, 113] based on the assumption of Gaussian measurement error. The weights are chosen as inversely proportional to the variances of measurement errors. Most of related work assumes that the variances of PMU measurements are fully-known or are chosen empirically, without addressing how the variances are quantified in practice. For instance in [114], the standard deviation of each measurement is defined as a linear function of the measurement value and the full scale value of instrumentation. Reference [19] extended it to a PMU-based state estimation algorithm. In another example of [24], the weights are both set to unity for the measurements of bus voltage magnitude and angle, is set to the smaller value between unity and the reciprocal of magnitude for the measurement of line current magnitude, and is set to 0.2 for the measurement of line current angle. Empirical weighting selections may perform well for specific applications, but can be difficult to apply to general cases. Thus, we need a systematic mechanism to improve its efficiency and applicability when quantifying the measurement variance, which has rarely been addressed in literature. Over the last two decades reweighted or adjusted techniques [115, 116, 117, 118] are introduced and integrated into WLS regressions for PSSE, with the aim to deal with non-Gaussian distributions and dependent measurements [119], or to obtain an efficient estimation scheme with certain robustness to outliers [120]. The work in [115] proposes an iterative re-weighted least square method where given rotations are implemented to solve a non-quadratic optimization problem, while the work 97



Detrended data

SEL-421 Protection Relays and PMU · · ·

Relay A/D Phasor estiamtor

PMU connection tester

SEL-PDC-5073 3

3

Collect data





Read data locally

98

SEL-421 Protection Relays and PMU

Opal-RT real-time simulator 3 signal streams

3

· · ·

Relay A/D Phasor estiamtor

SEL-PDC-5073 3



Collect and store data for analysis

3-phase grid signal

Quantifying PMU measurement weights

Instrument transformers CT/PT

PMU

PDC

Communication channel & update

PMU-based state estimation

Figure 7.1: Synchrophasor measurement chain: blue boxes are devices that may affect the measurement uncertainty; the relation between the input and output of the red box under different noise levels is the main interest herein.

in [116] uses a similar scheme to integrate weighted least absolute value (WLAV) in PSSE. An algorithm is proposed in [117] that adjusts the covariance matrix to take into account of the network parameter uncertainty. Furthermore, a non-diagonal covariance matrix is used in [118] to overcome measurement dependencies appearing in WLS regressions. On the other hand, an adaptive scheme is proposed in [121] to update the variances of the measurement noises, based on the calculated residuals over several past measurement snapshots. This result is further extended in [122] to improve its computational efficiency and resilience against measurement redundancy. Although using the aforementioned recursive procedures to update measurement covariance is suitable for on-line implementation, most of them still requires an initialization procedure. Thus, a systematic mechanism to quantify the measurement noise variances will benefit not only the classic WLS algorithm but also the adaptive algorithms mentioned above. Recently published paper [123] fills in this blank by performing theoretical analysis of uncertainty associated with the chain of involved devices. It concludes with an algebraic expressions of measurement uncertainty contributed by three major sources of errors, i.e., instrument transformers, measurement devices, and deadband. However, this analysis is conducted for the conventional measurement chain containing remote terminal units (RTUs) and associated devices, which have major differences from those used for the synchrophasor measurement chain. This chapter focuses on the role that a PMU plays in the synchrophasor measurement chain. We quantify PMU measurement uncertainty resulting from phasor calculation algorithms and the device’s internal filtering. Particularly, we examine the PMU outputs’ variances and expected values given different input variances. This procedure can be represented by the red box in the synchrophasor measurement chain shown in Fig. 7.1. In this chapter we will explore two approaches [124] to quantify PMU measurements: off-line simulation and hardware-in-the-loop (HIL) simulation. The off-line simulation characterizes the statistical relation between the inputs and outputs of PMUs for various input signals and the HIL simulation evaluates the impact of including actual hardware of PMUs.

3 signal

noise

3 signal with harmonics



Unbalanced 3 signal

Reference PMU Sequence Analyzer

Calculate the mean and standard deviation

0

50Hz

50*32 Hz

A teaching tool for phasor measurement estimation



Calculate the mean and standard deviation

7.1. Quantification approaches

99

ection r

Simulink/Matlab

ta

ection

a

Perfect 3 signal

Gaussian noise

3 signal with harmonics



Unbalanced 3 signal

50*32 Hz

Reference PMU Sequence Analyzer

  0

Calculate the mean and standard deviation

50Hz A teaching tool for phasor measurement estimation



Calculate the mean and standard deviation

Figure 7.2: Set-up for the off-line simulations

7.1

Quantification approaches

As indicated by (5.2) in Chapter 5, quantifying weights in WLS is transformed to quantifying the variances of the modeling uncertainty and measurement noise. In this thesis, we focus on quantifying measurement noise. Evaluating the network model uncertainties remains part of the future work. Before developing methods to quantify these variances, the following questions arise: How is the variance of a PMU output affected given different inputs noises? How sensitive are the output magnitude and angle w.r.t varying input variances? And how these would affect the choice of weightings in the WLS algorithm? Furthermore, how these answers would change if the system signal is of low quality, such as mixed with harmonics or unbalanced in three phases? These questions are firstly studied through off-line simulation. Off-line simulation provides a basic relationship between the input’s variance and the output’s expected value and variance. Furthermore, to evaluate the influences of actual PMU hardware, an investigation is conducted through the second quantification approach, using HIL simulation, where a physical PMU is included in the simulation loop.

7.1.1

Off-line simulation

The off-line simulation focuses on evaluating the performance of the PMU’s output when changing the variance of the input signal. It is conducted in matlab/Simulink and the set-up is shown in Fig. 7.2. The three red blocks represent three test scenarios: (i) perfect three-phase signal with the amplitude of 1; (ii) third-order harmonic with 0.5 gain on each phase signal; and (iii) unbalanced three-phase signal with the amplitude of 1, 0.95, 1, respectively. Three test scenarios are also described in Table 7.1. The signal frequency for all the three scenarios is 50 Hz and sampled at 1.6 kHz (i.e. 32 samples per cycle). After the three-phase signal is generated, Gaussian noise is added to each phase with the same variance but different initial seeds. For each scenario different input noise variances are assigned, i.e., [0, 0.001, 0.025, 0.05, 0.075, 0.1, 0.2, 0.5]. The

100

Quantifying PMU measurement weights Table 7.1: Three test scenarios Scenario I

Perfect three-phase signal

Scenario II

Third-order harmonic with 0.5 gain is added to each phase

Scenario III

Unbalanced three-phase signal having the amplitude of 1, 0.95 and 1

sampling frequency of the Gaussian noises is 1.6 kHz . Then the three-phase signal is sent to the PMU model. Two PMU models are used in this step: (i) the sequence analyzer block [125] from matlab/Simulink, whose function is to compute the positive-, negative-, and zero-sequence components of a three-phase signal; (ii) a teaching tool for phasor measurement estimation proposed by [126]. The first model is taken as a reference PMU herein. Please note that the PMU models in the off-line simulation only represent the phasor calculator segment rather than a whole physical PMU model. The last step is to store two PMUs’ outputs and calculate their expected values and variances for both magnitude and angle. After organizing all these results, it is explained how the PMU outputs change given different input variances for all the three scenarios.

7.1.2

Hardware-in-the-loop simulation

Compared to the off-line simulation, HIL simulation introduces a physical PMU into the simulation loop with the purpose of imitating a field test. As the inputs of physical PMUs need to be real-time analog signals, a real-time simulator, Opal-RT’s eMegaSim, in the SmarTS Lab [127] at KTH is used to generate the input signals for the PMU. The set-up for the HIL simulation is shown in Fig. 7.3. It starts with developing a three-phase signal generation model in RT-LAB, a software that acts as the interface between the Opal-RT real-time simulator and the users. After compiling and loading the model into the real-time simulator, it runs with a real-time clock. The output of the simulator is a three-phase analog signal and it is sent to SEL421 Protection Relay and PMU. Note that PMUs installed in real power systems measure high voltage and current phasors through instrument transformers at low voltage and current levels (±300V , 0-5A), however the real-time simulator can not generate such values. Therefore the simulator’s outputs are sent to the low-level interface connections inside the PMU, which has a limit of 2.33 V rms [128]. Then the phasors calculated by the PMU is sent to the phasor data concentrator (PDC), SEL-PDC-5073, whose function is to receive and time-synchronize phasor data from multiple PMUs to produce a real-time, time-aligned output data stream. Because these experiments were carried out in a controlled environment, where loss of data and latency could be controlled, it is reasonable to neglect the possible influences of both the PDC and the communication channels. Therefore, for practical

3 signal generation

SEL-421 Protection Relays and PMU

Opal-RT real-time simulator

RT-Lab 3

3 signal streams

3

· · ·

Relay A/D Phasor estiamtor

PMU connection tester

SEL-PDC-5073 3

3

Collect data



Read data locally



7.2. Results

101 SEL-421 Protection Relays and PMU

Opal-RT real-time simulator

RT-Lab 3 signal generation

3

3 signal streams

3

· · ·

Relay A/D Phasor estiamtor

SEL-PDC-5073 3



Collect and store data for analysis

Figure 7.3: Set-up for the HIL simulation experiments purposes, it is assumed that the PDC and the communication channels have no effect on the phasor computation, neither on the measurement uncertainty in our study case. The influence of hardware on the measurement uncertainty is only associated with the PMU.

7.2 7.2.1

Results Off-line simulation

Off-line simulation is conducted for three scenarios as introduced in Subsection 7.1.1, and its results are shown in Figs. 7.4, 7.5 and 7.6. Figure 7.4 shows the variances and the expected values of the PMU outputs when varying the input’s variances for Scenario I. Figure 7.4a depicts the variances of the output magnitude and angle. For the range of input σ 2 ∈ [0 0.5], the figure indicates that both variances for magnitude and angle change linearly along with the input variance. In order to verify this observation, we firstly apply the “fit" function [129] and the “poly1" [130] model to fit the curves in Fig. 7.4a. The fitting results are straighting lines passing through the origin, whose expressions are presented in Table 7.2. Then these results are evaluated by the normalized root mean square error (NRMSE), which is defined as q P n 1 2 i=1 (yˆi − yi ) RMSE n NRMSE = = , (7.1) |ymax − ymin | |ymax − ymin | where yi is the value of ith data; yˆi is the corresponding value for yi on the fitted curve; ymax and ymin are the maximum and minimum values among all the data. By taking account of the range of the data, NRMSE can help to compare the RMSEs among different quantities. In this sense, NRMSE becomes a global evaluation index used for all the three scenarios. The NRMSE for both variances of magnitude and angle in Fig. 7.4a are approximately in the order of 0.1% to 0.2%, which are sufficiently small. Hence, the fitted

102

Quantifying PMU measurement weights

Magnitude (volt)

Magnitude (volt) 1.007

refPMU PMU

0.01

µ of the output

σ2 of the output

0.012

0.008 0.006 0.004 0.002 0

0

0.1 2

0.2

0.3

0.4

1.005 1.004 1.003 1.002 1.001 1

0.5

refPMU PMU

1.006

σ of the input noise

0

0.1

Angle (deg.)

0.4

0.5

0

30

refPMU PMU

25

µ of the output

σ2 of the output

0.3

Angle (deg.)

35

20 15 10 5 0

0.2

σ2 of the input noise

0

0.1

0.2

0.3

0.4

σ2 of the input noise (a)

0.5

refPMU PMU

−0.05 −0.1 −0.15 −0.2 −0.25 0

0.1

0.2

0.3

0.4

0.5

σ2 of the input noise (b)

Figure 7.4: PMU outputs with increasing input noise for Scenario I: Perfect three-phase signal. (7.4a) (two left figures): covariances of the output magnitude (top) and angle (bottom). (7.4b) (two right figures): expected values of the output magnitude (top) and angle (bottom).

7.2. Results

103

Table 7.2: Off-line simulation results Scenario I

Fitted Curve

NRMSE

Type

0.00115

linear

y = 0.02257x + 1.252 × 10

0.00110

linear

y = 69.19x − 0.08718

0.000983

linear

y = 46.6x − 0.0516

0.00224

linear

µ|V | ref

y = 0.01116x + 1

0.00792

linear

µ|V |

y = 0.01176x + 1

0.00753

linear

2 σ|V | ref 2 σ|V | σθ2 ref σθ2

y = 0.0222x + 1.281 × 10

−5 −5

y = −0.3249x

0.5471

− 0.00372

0.00145

quasi square root

y = −0.3157x

0.5362

− 0.002976

0.00120

quasi square root

y = 0.02156x + 6.127 × 10−6

0.000585

linear

2 σ|V |

y = 0.0218x + 6.128 × 10−6

0.000574

linear

σθ2 ref σθ2

y = 66.14x − 0.03225

0.00103

linear

y = 41.49x − 0.01667

0.000877

linear

µ|V | ref

y = 0.0138x + 1

0.0309

linear

µ|V |

y = 0.01446x + 1

0.0298

linear

µθ ref µθ Scenario II 2 σ|V |

ref

µθ ref µθ

y = −0.5195x

0.5072

− 0.00129

0.000351

quasi square root

y = −0.4908x

0.5008

− 0.000448

0.000201

quasi square root

0.000601

linear

0.000593

linear

y = 68.43x − 0.03446

0.00106

linear

Scenario III 2 σ|V |

ref

y = 0.02156x + 6.298 × 10−6 −6

2 σ|V | σθ2 ref σθ2

y = 0.0217x + 6.319 × 10 y = 42.92x − 0.01792

0.000913

linear

µ|V | ref

y = 0.01397x + 0.9837

0.0306

linear

µ|V |

y = 0.01464x + 0.9837

0.0295

linear

µθ ref µθ

y = −0.5285x

0.5074

y = −0.4991x

0.501

− 0.00134

0.000361

quasi square root

− 0.000492

0.000209

quasi square root

104

Quantifying PMU measurement weights

curves are valid. Furthermore, it implies the weighting matrix generated using a set of variances under the same noise level will be valid for most noise levels. On the other hand, Fig. 7.4a shows that the angle variance changes dramatically compared to the magnitude variance, which can also be observed from the slopes presented in Table 7.2. This indicates that the angle derivation inside the phasor estimators are more sensitive to the inputs noises. Moreover, under the same input noise, different instrument models (refPMU and PMU phasor estimators) have different impacts on angle variance. As a result, different weights are needed when different instrument models are employed. When the variance of the input noise is increased from 0 to 0.5, the expected values of both the magnitude and angle in the output deviate from their initial values, which although is much less than the derivation in the variances. In Fig. 7.4b, it is observed the expected values of the output magnitudes are linear, however, when using the “poly1" model to fit these curves, their NRMSEs are in the order of 0.7%, which is larger than the values for variances. This is due to the slight nonlinearity in the beginning of the curves when the input variances are small. In contrast, the expected value of the output angle is clearly nonlinear, with a similar appearance of a negative square root function. Hence, “power2" [130] model is applied to fit the curves and the calculated NRMSEs are in the order of 0.1% to 0.2%, which are same with the values for variances. Figures 7.5 and 7.6 show the variances and the expected values of PMU outputs when varying the input’s variances for Scenario II and III, respectively. It can be seen from Fig. 7.5 that introducing harmonics only slightly affects the variances of the magnitude and the angle. However, it increases the absolute expected values of both magnitude and angle. Similarly, as shown in Figure 7.6, introducing unbalanced three-phase signal affects mostly the absolute expected values of both magnitude and angle. Therefore, a plausible conclusion is harmonics or unbalanced three-phase conditions will mainly affect the expected values of output magnitude and angle. In summary, the main observations from comparing Figs. 7.4-7.6 and Table 7.2 are 2 2 2 • σ|V | /σθ is constant ∀σinput ∈ [0 0.5]. Therefore, the weighting matrix generated using a set of variances under the same noise level will be valid for most noise levels. 2 2 2 • σ|V |  σθ , ∀σinput ∈ [0 0.5]. This implies that angle calculation inside phasor estimators are more sensitive to the inputs noises.

• σθ2 (refPMU) > σθ2 (PMU). It indicates that under the same input noise different instrument models (refPMU and PMU phasor estimators) have different impacts on angle variance. • Harmonics or unbalanced three-phase conditions mainly affect the expected values of output magnitude and angle.

7.2. Results

105

Magnitude (volt) 0.01 0.008

Magnitude (volt) 1.01

refPMU refPMU with harmonics PMU PMU with harmonics

µ of the output

σ2 of the output

0.012

0.006 0.004 0.002 0

0

0.1 2

0.2

0.3

0.4

1.008 1.006 1.004 1.002 1

0.5

0

σ of the input noise

µ of the output

20 15 10 5 0

0

0.1 2

0.2

0.3

0.4

σ of the input noise (a)

0.3

0.4

0.5

Angle (deg.)

2

σ of the output

25

0.2

0

refPMU refPMU with harmonics PMU PMU with harmonics

30

0.1

σ2 of the input noise

Angle (deg.) 35

refPMU refPMU with harmonics PMU PMU with harmonics

0.5

−0.1

−0.2

−0.3

−0.4 0

refPMU refPMU with harmonics PMU PMU with harmonics 0.1 0.2 0.3 2

0.4

0.5

σ of the input noise (b)

Figure 7.5: PMU outputs with increasing input noise for Scenario II: A third harmonics with 0.5 gain is added to each phase. (7.5a) (two left figures): covariances of the output magnitude (top) and angle (bottom). (7.5b) (two right figures): expected values of the output magnitude (top) and angle (bottom).

106

Quantifying PMU measurement weights Magnitude (volt)

Magnitude (volt) 0.01 0.008

1.015 refPMU refPMU unbalanced 3Φ PMU PMU unbalanced 3Φ

µ of the output

σ2 of the output

0.012

0.006 0.004 0.002 0

0

0.1 2

0.2

0.3

0.4

1.01 1.005

refPMU refPMU unbalanced 3Φ PMU PMU unbalanced 3Φ

1 0.995 0.99 0.985 0.98 0

0.5

σ of the input noise

0.1

Angle (deg.) 30 25

15 10 5 0

0.1 2

0.2

0.3

0.4

0.5

Angle (deg.)

20

0

0.3

0

refPMU refPMU unbalanced 3Φ PMU PMU unbalanced 3Φ

µ of the output

σ2 of the output

35

0.2

σ2 of the input noise

0.4

σ of the input noise

0.5

−0.1

−0.2

−0.3

−0.4 0

(a)

refPMU refPMU unbalanced 3Φ PMU PMU unbalanced 3Φ 0.1

0.2

0.3

0.4

0.5

σ2 of the input noise

(b)

Figure 7.6: PMU outputs with increasing input noise for Scenario III: Unbalanced threephase signal with the magnitude of 1, 0.95 and 1. (7.6a) (two left figures): covariances of the output magnitude (top) and angle (bottom). (7.6b) (two right figures): expected values of the output magnitude (top) and angle (bottom).

7.2.2

Hardware-in-the-loop simulation

The results of HIL simulation are shown in Fig. 7.7 and Table 7.3. Note that two y-axes are applied for Fig. 7.7a, where it can be observed the significant decrease in the output’s variances due to the PMU’s filtering stage. The PMU includes different low-pass filters: one installed before the A/D conversion, and two after the calculation of real and complex components, respectively [131]. The PMU filters also affect the expected values of the magnitude and angle in the outputs. In order to evaluate these influences, the IEEE Standard for Synchrophasor Measurements for Power Systems is utilized here. Specifically, the requirement for the maximum total vector error (TVE) of the steady-state synchrophasor measurement is 1%. This 1% criterion can be visualized as a small circle drawn on the end of the

10-4 2 1.5

0.01 1 offline0.5 HIL 0 0.3

0.4

0.8 0.6

20

0.4

10

offline0.2 HIL

0

0.1 2

0.2

0.3

0.4

0 0.5

of the input noise

(a)

0.1 2

30

0

offline HIL

0.99

of the input noise Angle (deg.)

40

Magnitude (volt)

1

0.97 0

0 0.5

2

2

of the output

2

0.2

of the output

0.1

of the output

0

1.01

0.98

2

0.005

of the output

Magnitude (volt)

of the output

0.015

107

2

of the output

7.2. Results

0.2

0.3

0.4

0.5

of the input noise Angle (deg.)

0.1 0

offline HIL

-0.1 -0.2 -0.3 0

0.1 2

0.2

0.3

0.4

0.5

of the input noise

(b)

Figure 7.7: PMU outputs with increasing input noise for HIL simulation. (7.7a) (leftcolumn figures): covariances of the output magnitude (top) and angle (bottom) when varying the variance of input noise. (7.7b) (right-column figures): expected values of the output magnitude (top) and angle (bottom) when varying the variance of input noise. phasor. The maximum magnitude error is 1% when the error in phase is zero, and the maximum error in angle is just under 0.573◦ when the error in magnitude is zero [131]. The objective is to determine the noise level under which the output will violate the 1% criterion. • Regarding the output magnitude, in order to meet the 1% criterion, the condition |µ − 1| + σ ≤ 0.01 has to be satisfied. Since the expected value of the magnitude is smaller than 1 (due to the PMU’s filtering), the input signals will not violate the 1% criterion when the output magnitude satisfies µ − σ ≥ 0.99. As shown in Table 7.3, this criterion is met when the variance of the input noise is between 0.1 and 0.2. • Similarly, regarding the output angle, in order to meet the 1% criterion, |µ−0|+σ ≤ 0.537. Because the expected value is larger than 0, the input signals

Quantifying PMU measurement weights 108

2 σ|V |

2 σinput

0.998354

0.00045272

(8.574 77 × 10−6 )2

0

0.0290496

0.998581

0.105422

0.001910812

0.01

0.0283797

0.998614

0.1707882

0.003058572

0.025

0.0244459

0.998696

0.246782

0.00433252

0.05

0.0314455

0.998662

0.2941442

0.005257092

0.075

0.0270704

0.998645

0.3437342

0.005994112

0.1

0.0464034

0.997086

0.4656662

0.008182912

0.2

0.979782

0.7735982

0.0127052

0.5

Table 7.3: HIL simulation results

σθ2 0.0282645

0.0395692

µθ

µ|V |

7.3. Summary

109

will not violate the 1% criterion when the output angle satisfies µ + σ ≤ 0.537. 2 As shown in Table 7.3, this criterion is met when σinput ≈ 0.2. • In summary, based on the PMU used in this HIL simulation, injecting signals 2 with σinput ≥ 0.2 may violate the 1% criterion. Nevertheless, 0.1 is already a relatively large number for variances. As it is shown in Fig. 7.7b, the expected values for both magnitude and angle remain considerably unchanged when 2 σinput ≤ 0.1. The off-line simulation studies in Section 7.2.1 indicate that the measurement’s variance quantified around the operating point holds for most noise levels. Furthermore, 2 the HIL simulation shows that the input noise needs to be bounded to σinput ≤ 0.1.

7.3

Summary

Two approaches to quantify the PMU measurement weights are carried out in this chapter: off-line simulation and HIL simulation. First, in off-line simulation we demonstrate how the expected value and variance of the PMU measurement change given inputs with different variances. This is carried out using PMU models described in [125] and [126]. Different scenarios are analyzed, including the perfect three-phase signals, signals with harmonics, and unbalanced three-phase signals. A linear relationship between the input’s and output’s variances implies that the PMU measurement variance quantified at any normal operating condition is capable to generate a weighting matrix that is valid for most noise levels. In addition, angle derivation inside the phasor estimators is more sensitive to the input’s noises compared to magnitude. These results are ideal as different scenarios are applied to the PMU models that do not include the impact of hardware. Therefore, we examine the influence of actual hardware on the previous results that are based purely on simulation, via hardware-in-the-loop (HIL) simulation.

Chapter 8

Conclusion At the beginning of this thesis, several practical examples were used to motivate the need of fast and accurate power system state estimation techniques to facilitate meeting the challenges of increasing penetration of renewable energy sources and how to exploit them through HVDC and power electronic-based devices. In particular, the feasibility of PMU-only state estimation is highlighted as a viable path forward as the PMU installation and deployment is dramatically increasing. In addition, the emphasis is laid on the use of HVDC and other power electronic systems, which require a state estimation solution for hybrid AC/DC grids. These motivation examples were followed by an overview on power system state estimation techniques in Chapter 2. The key roles that a state estimation plays in a power system and the development of state estimation schemes when using PMUs and considering power electronic devices were presented. To meet the diverse needs for state estimation, measurements over consecutive instants or a system model that considers time evolution can be utilized in addition to the static measurement model, leading to the so-called forecasting-aided state estimation and dynamic state estimation, respectively. Furthermore, state estimation is not exclusive for transmission systems any more, but becomes applicable to distribution systems as well. The architecture of state estimations, i.e., models, formulation and computation, is no longer confined to a centralized paradigm; distributed or hierarchical schemes may be a better choice in some cases. For each of the aforementioned aspects, some related research in literature was discussed. Chapter 3 focused on the formulation and derivation of state estimation methods, particularly the formulation and solution of the conventional state estimation approach. In addition, the test systems that were developed and implemented for case studies were introduced. These contents provided a background for the following chapters. Chapter 4 and 5 developed a paradigm of using PMU data to solve static state estimation for hybrid AC/DC grids, but using different problem formulations and network models. Chapter 4 attempted a linear scheme where the measurement model was linear and the linear WLS algorithm was applied for solution. In addition, linear 111

112

Conclusion

network models for AC transmission network and classic HVDC link were developed. Although linear WLS can be solved directly (i.e., without iterations), it conceals the state variables to be in rectangular coordinates, which loses the advantage of angle bias detection and correction. Therefore, in Chapter 5, state variables in polar coordinates were used, leading to a nonlinear problem formulation. The nonlinear WLS algorithm was applied for its solution. At the same time, we proposed a novel measurement model that separates the errors due to modeling uncertainty and measurement noise so that different weights can be assigned to them separately. Nonlinear network models for AC transmission network, classic HVDC link, voltage source converter (VSC)-HVDC, and FACTS were developed and tested. The case studies in Chapter 4 and 5 showed that both linear and nonlinear solution schemes performed adequately when the system was under steady-state or quasi-steady state, but less satisfactorily when the system was under large dynamic changes and power electronic devices react to these changes. This implies that additional modeling details need to be included to obtain higher accuracy during system transient dynamics. On the other hand, most dynamic state estimators and forecasting-aided state estimators are computationally demanding. In Chapter 6, we proposed a pseudo-dynamic modeling approach that can improve the estimation accuracy during transients without significantly increasing the estimation’s computational burden. It maintains the formulation and algorithm used for the nonlinear state estimation in Chapter 5 and only adds difference equations from dynamic models of certain components to the original static network equations. This approach greatly reduces the workload of re-composing network models when updating existing algorithms. To illustrate this approach, pseudodynamic network models for static synchronous compensator (STATCOM), as an example of a FACTS device, and VSC-HVDC link were developed and validated via simulation. Last but not least, in Chapter 7, we proposed two approaches to quantify PMU measurement weights: off-line simulation and hardware-in-the-loop (HIL) simulation. The findings provide better guidance for selecting proper weights for power system state estimation.

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